SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

</head>

Out(Fn)

</h1>

</header>

In <a href="Mathematics" title="Mathematics">mathematics</a>, Out(Fn) is the <a href="Outer_automorphism_group" title="Outer automorphism group">outer automorphism group</a> of a <a href="Free_group" title="Free group">free group</a> on n <a href="Generating_set_of_a_group" title="Generating set of a group">generators</a>. These groups are at universal stage in <a href="Geometric_group_theory" title="Geometric group theory">geometric group theory</a>, as they act on the set of <a href="Presentation_of_a_group" title="Presentation of a group">presentations</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> generators of any <a href="Finitely_generated_group" title="Finitely generated group">finitely generated group</a>.<a href="#cite_note-1">[1]</a> Despite geometric analogies with general <a href="General_linear_group" title="General linear group">linear groups</a> and <a href="Mapping_class_group_of_a_surface" title="Mapping class group of a surface">mapping class groups</a>, their complexity is generally regarded as more challenging, which has fueled the development of new techniques in the field.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> be the free nonabelian group of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

</semantics>

</math><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy">. The set of <a href="Inner_automorphism" title="Inner automorphism">inner automorphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy">, i.e. automorphisms obtained as conjugations by an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy">, is a <a href="Normal_subgroup" title="Normal subgroup">normal subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Inn} (F_{n})\triangleleft \mathrm {Aut} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Inn} (F_{n})\triangleleft \mathrm {Aut} (F_{n})}</annotation>

</semantics>

</math><img src="./7e4a6d5e1c215bec624cd5231172c3c5b6805bd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.604ex; height:2.843ex;" alt="{\displaystyle \mathrm {Inn} (F_{n})\triangleleft \mathrm {Aut} (F_{n})}" loading="lazy">. The outer automorphism group of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> is the quotient<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n}):=\mathrm {Aut} (F_{n})/\mathrm {Inn} (F_{n}).}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n}):=\mathrm {Aut} (F_{n})/\mathrm {Inn} (F_{n}).}</annotation>

</semantics>

</math>An element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> is called an outer class.

<div class="mw-heading mw-heading2"><h2 id="Relations_to_other_groups">Relations to other groups</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Linear_groups">Linear groups</h3></div>

The <a href="Abelianization" class="mw-redirect" title="Abelianization">abelianization</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\to \mathbb {Z} ^{n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}\to \mathbb {Z} ^{n}}</annotation>

</semantics>

</math><img src="./f05c7eb703995874397dc7e4284691651220e281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.096ex; height:2.676ex;" alt="{\displaystyle F_{n}\to \mathbb {Z} ^{n}}" loading="lazy"> induces a <a href="Homomorphism" title="Homomorphism">homomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> to the general linear group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (n,\mathbb {Z} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n,\mathbb {Z} )}</annotation>

</semantics>

</math><img src="./8214d11442717fc11dd8f42a6363061cfe9dade7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n,\mathbb {Z} )}" loading="lazy">, the latter being the <a href="Automorphism_group" title="Automorphism group">automorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation>

</semantics>

</math><img src="./a9b5de7ced4588982b574fe19894aec6a3ca4c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}" loading="lazy">. This map is onto, making <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> a <a href="Group_extension" title="Group extension">group extension</a>,

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\to \mathrm {Tor} (F_{n})\to \mathrm {Out} (F_{n})\to \mathrm {GL} (n,\mathbb {Z} )\to 1}</annotation>

</semantics>

</math><img src="./ccf1eb96b9b1e1b764b2fe791b10790c06c40fa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.649ex; height:2.843ex;" alt="{\displaystyle 1\to \mathrm {Tor} (F_{n})\to \mathrm {Out} (F_{n})\to \mathrm {GL} (n,\mathbb {Z} )\to 1}" loading="lazy">.</dd></dl>

The kernel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Tor} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Tor} (F_{n})}</annotation>

</semantics>

</math><img src="./e07bfc565ffcaf5f14a7c1945006d71eef96a095.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.274ex; height:2.843ex;" alt="{\displaystyle \mathrm {Tor} (F_{n})}" loading="lazy"> is the <a href="Torelli_group" class="mw-redirect" title="Torelli group">Torelli group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy">.

The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{2})\to \mathrm {GL} (2,\mathbb {Z} )}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{2})\to \mathrm {GL} (2,\mathbb {Z} )}</annotation>

</semantics>

</math><img src="./8c520ad286097443eb7b8918eac4def0925682d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.811ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{2})\to \mathrm {GL} (2,\mathbb {Z} )}" loading="lazy"> is an <a href="Isomorphism" title="Isomorphism">isomorphism</a>. This no longer holds for higher ranks: the Torelli group of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{3}}</annotation>

</semantics>

</math><img src="./afd489b4f91358d1094a63dec2abf9699478e6c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{3}}" loading="lazy"> contains the automorphism fixing two basis elements and multiplying the remaining one by the commutator of the two others.

By definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Aut} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Aut} (F_{n})}</annotation>

</semantics>

</math><img src="./ef32a67573dc3e1d5473f953d2e3ffc4333da1f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.463ex; height:2.843ex;" alt="{\displaystyle \mathrm {Aut} (F_{n})}" loading="lazy"> is an extension of the inner automorphism group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Inn} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Inn} (F_{n})}</annotation>

</semantics>

</math><img src="./a0a15400542b40906f9bbbcd7c601adb7e0817d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.947ex; height:2.843ex;" alt="{\displaystyle \mathrm {Inn} (F_{n})}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy">. The inner automorphism group itself is the image of the <a href="Conjugacy_class#Conjugacy_as_group_action" title="Conjugacy class">action by conjugation</a>, which has kernel the <a href="Center_(group_theory)" title="Center (group theory)">center</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(F_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(F_{n})}</annotation>

</semantics>

</math><img src="./f8a899eba32c4cc879310e246cc7beb794c021c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.203ex; height:2.843ex;" alt="{\displaystyle Z(F_{n})}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(F_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(F_{n})}</annotation>

</semantics>

</math><img src="./f8a899eba32c4cc879310e246cc7beb794c021c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.203ex; height:2.843ex;" alt="{\displaystyle Z(F_{n})}" loading="lazy"> is trivial for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>

</semantics>

</math><img src="./e6bf67f9d06ca3af619657f8d20ee1322da77174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}" loading="lazy">, this gives a short exact sequence<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\rightarrow F_{n}\rightarrow \mathrm {Aut} (F_{n})\rightarrow \mathrm {Out} (F_{n})\rightarrow 1.}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\rightarrow F_{n}\rightarrow \mathrm {Aut} (F_{n})\rightarrow \mathrm {Out} (F_{n})\rightarrow 1.}</annotation>

</semantics>

</math>For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>

</semantics>

</math><img src="./e6bf67f9d06ca3af619657f8d20ee1322da77174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}" loading="lazy">, there are embeddings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Aut} (F_{n})\longrightarrow \mathrm {Out} (F_{n+1})}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Aut} (F_{n})\longrightarrow \mathrm {Out} (F_{n+1})}</annotation>

</semantics>

</math><img src="./1f634f263c80da1acec2e3c8c23c65bae03143ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.187ex; height:2.843ex;" alt="{\displaystyle \mathrm {Aut} (F_{n})\longrightarrow \mathrm {Out} (F_{n+1})}" loading="lazy"> obtained by taking the outer class of the extension of an automorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> fixing the additional generator. Therefore, when studying properties that are inherited by subgroups and quotients, the theories of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Aut} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Aut} (F_{n})}</annotation>

</semantics>

</math><img src="./ef32a67573dc3e1d5473f953d2e3ffc4333da1f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.463ex; height:2.843ex;" alt="{\displaystyle \mathrm {Aut} (F_{n})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> are essentially the same.

<div class="mw-heading mw-heading3"><h3 id="Mapping_class_groups_of_surfaces">Mapping class groups of surfaces</h3></div>

Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> is the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of a <a href="Bouquet_of_circles" class="mw-redirect" title="Bouquet of circles">bouquet of n circles</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> can be described topologically as the <a href="Mapping_class_group" title="Mapping class group">mapping class group</a> of a bouquet of n circles (in the <a href="Homotopy_category" title="Homotopy category">homotopy category</a>), in analogy to the mapping class group of a closed <a href="Surface_(mathematics)" title="Surface (mathematics)">surface</a> which is isomorphic to the outer automorphism group of the fundamental group of that surface.

Given any finite graph with fundamental group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy">, the graph can be "thickened" to a surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> with one boundary component that retracts onto the graph. The <a href="Mapping_class_group_of_a_surface#The_Birman_exact_sequence" title="Mapping class group of a surface">Birman exact sequence</a> yields a map from the mapping class group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MCG} (S)\longrightarrow \mathrm {Out} (F_{n})}">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {MCG} (S)\longrightarrow \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./9fc60b4bc07a60da93f3899f256a0fad0664be5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.566ex; height:2.843ex;" alt="{\displaystyle \mathrm {MCG} (S)\longrightarrow \mathrm {Out} (F_{n})}" loading="lazy">. The elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> that are in the image of such a map are called geometric. Such outer classes must leave invariant the cyclic word corresponding to the boundary, hence there are many non-geometric outer classes. A converse is true under some irreducibility assumptions,<a href="#cite_note-2">[2]</a> providing geometric realization for outer classes fixing a conjugacy class.

<div class="mw-heading mw-heading2"><h2 id="Known_results">Known results</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\geq 4}</annotation>

</semantics>

</math><img src="./5ca22e7b7ea52104f9002a9f581e1953b3b7aa09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\textstyle n\geq 4}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./d930820a717176bb1ef24f97f13a49c6a0ae8d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (F_{n})}" loading="lazy"> is not linear, i.e. it has no <a href="Faithful_representation" title="Faithful representation">faithful representation</a> by matrices over a field (Formanek, Procesi, 1992);<a href="#cite_note-3">[3]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

</semantics>

</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy">, the <a href="Dehn_function" title="Dehn function">isoperimetric function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> is exponential (Hatcher, Vogtmann, 1996);<a href="#cite_note-4">[4]</a></li>

<li>The <a href="Tits_alternative" title="Tits alternative">Tits Alternative</a> holds in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy">: each subgroup is either <a href="Virtually" title="Virtually">virtually</a> <a href="Solvable_group" title="Solvable group">solvable</a> or else it contains a free group of rank 2 (Bestvina, Feighn, Handel, 2000);<a href="#cite_note-5">[5]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\geq 3}</annotation>

</semantics>

</math><img src="./d435515f4209d6aa1519189714c90d91d0b2737b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\textstyle n\geq 3}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (\mathrm {Out} (F_{n}))={1}}">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (\mathrm {Out} (F_{n}))={1}}</annotation>

</semantics>

</math><img src="./8e0e0dca3c2ff8d1ef6fe592146e2af4dd813579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.603ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (\mathrm {Out} (F_{n}))={1}}" loading="lazy"> (Bridson and Vogtmann, 2000);<a href="#cite_note-6">[6]</a></li>

<li>Every solvable subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> has a finitely generated <a href="Free_abelian_group" title="Free abelian group">free abelian</a> subgroup of finite index (Bestvina, Feighn, Handel, 2004);<a href="#cite_note-7">[7]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i>0}</annotation>

</semantics>

</math><img src="./1f49f2878fd68a89c3da37eb537198e887cf0293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i>0}" loading="lazy">, all but finitely many of the <a href="Group_cohomology#Group_homology" title="Group cohomology"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th-degree homology</a> morphisms induced by the sequence<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots \rightarrow \mathrm {Out} (F_{n-1})\rightarrow \mathrm {Out} (F_{n})\rightarrow \mathrm {Out} (F_{n+1})\rightarrow \ldots }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ldots \rightarrow \mathrm {Out} (F_{n-1})\rightarrow \mathrm {Out} (F_{n})\rightarrow \mathrm {Out} (F_{n+1})\rightarrow \ldots }</annotation>

</semantics>

</math> are isomorphisms (Hatcher and Vogtmann, 2004);<a href="#cite_note-8">[8]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\geq 2}</annotation>

</semantics>

</math><img src="./8a6e5c149b9241fd9451fcce9525c812fef16652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\textstyle n\geq 2}" loading="lazy">, the reduced <a href="C*-algebra" title="C*-algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C^{*}}</annotation>

</semantics>

</math><img src="./67bd07d7c9b2743c0a8bea9d2e82a7920b783f5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.176ex;" alt="{\textstyle C^{*}}" loading="lazy">-algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./d930820a717176bb1ef24f97f13a49c6a0ae8d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (F_{n})}" loading="lazy"> (i.e. the closure of its image under the <a href="Regular_representation" title="Regular representation">regular representation</a>) is simple;<a href="#cite_note-9">[9]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\geq 4}</annotation>

</semantics>

</math><img src="./5ca22e7b7ea52104f9002a9f581e1953b3b7aa09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\textstyle n\geq 4}" loading="lazy">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Gamma }</annotation>

</semantics>

</math><img src="./f677684faf84745068a3b602896ca59b0766be4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\textstyle \Gamma }" loading="lazy"> is a finite index subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./d930820a717176bb1ef24f97f13a49c6a0ae8d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (F_{n})}" loading="lazy">, then any subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./d930820a717176bb1ef24f97f13a49c6a0ae8d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (F_{n})}" loading="lazy"> isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Gamma }</annotation>

</semantics>

</math><img src="./f677684faf84745068a3b602896ca59b0766be4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\textstyle \Gamma }" loading="lazy"> is a conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Gamma }</annotation>

</semantics>

</math><img src="./f677684faf84745068a3b602896ca59b0766be4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\textstyle \Gamma }" loading="lazy"> (Farb and Handel, 2007);<a href="#cite_note-10">[10]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\geq 5}</annotation>

</semantics>

</math><img src="./5f64a4e7ebcd80506c52bd75f37fca0f43d45c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\textstyle n\geq 5}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./d930820a717176bb1ef24f97f13a49c6a0ae8d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (F_{n})}" loading="lazy"> has <a href="Kazhdan's_property_(T)" title="Kazhdan's property (T)">Kazhdan's property (T)</a> (Kaluba, Nowak, Ozawa, 2019 for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n=5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n=5}</annotation>

</semantics>

</math><img src="./b4a216f85733f1ae3781d5597adba09cda7a88a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\textstyle n=5}" loading="lazy">; Kaluba, Kielak, Nowak, 2021 for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 6}</annotation>

</semantics>

</math><img src="./154f45728701bd6df71c58558cdb2bdde36214ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 6}" loading="lazy">);<a href="#cite_note-11">[11]</a></li>

<li>Actions on <a href="Hyperbolic_metric_space" title="Hyperbolic metric space">hyperbolic</a> complexes satisfying <a href="Acylindrically_hyperbolic_group" title="Acylindrically hyperbolic group">acylindricity</a> conditions were constructed, in analogy with complexes like the <a href="Curve_complex" title="Curve complex">complex of curves</a> for mapping class groups;<a href="#cite_note-12">[12]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\geq 3}</annotation>

</semantics>

</math><img src="./d435515f4209d6aa1519189714c90d91d0b2737b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\textstyle n\geq 3}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./d930820a717176bb1ef24f97f13a49c6a0ae8d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\textstyle \mathrm {Out} (F_{n})}" loading="lazy"> is rigid with respect to measure equivalence (Guirardel and Horbez, 2021 preprint).<a href="#cite_note-13">[13]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Outer_space">Outer space</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Outer_space_(mathematics)" title="Outer space (mathematics)">Outer space (mathematics)</a></div>

Out(Fn) acts <a href="Geometric_group_action" title="Geometric group action">geometrically</a> on a <a href="Cell_complex" class="mw-redirect" title="Cell complex">cell complex</a> known as <a href="Marc_Culler" title="Marc Culler">Culler</a>–<a href="Karen_Vogtmann" title="Karen Vogtmann">Vogtmann</a> Outer space, which can be thought of as the <a href="Teichm%C3%BCller_space" title="Teichmüller space">Fricke-Teichmüller space</a> for a <a href="Bouquet_of_circles" class="mw-redirect" title="Bouquet of circles">bouquet of circles</a>.

<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

A point of the outer space is essentially an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free <a href="Homotopy" title="Homotopy">homotopy</a> class of a <a href="Homotopy#Homotopy_equivalence_and_null-homotopy" title="Homotopy">homotopy equivalence</a> from X to the bouquet of n circles. An <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-graph is just a weighted <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> with weights in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3.

A more descriptive view avoiding the homotopy equivalence f is the following. We may fix an identification of the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of the bouquet of n circles with the <a href="Free_group" title="Free group">free group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> in n variables. Furthermore, we may choose a <a href="Spanning_tree_(mathematics)" class="mw-redirect" title="Spanning tree (mathematics)">maximal tree</a> in X and choose for each remaining edge a direction. We will now assign to each remaining edge e a word in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> in the following way. Consider the closed path starting with e and then going back to the origin of e in the maximal tree. Composing this path with f we get a closed path in a bouquet of n circles and hence an element in its fundamental group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy">. This element is not well defined; if we change f by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique <a href="Free_group#Construction" title="Free group">cyclically reduced</a> element in this conjugacy class. It is possible to reconstruct the free homotopy type of f from these data. This view has the advantage, that it avoids the extra choice of f and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.

The operation of Out(Fn) on the outer space is defined as follows. Every automorphism g of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> induces a self homotopy equivalence g′ of the bouquet of n circles. Composing f with g′ gives the desired action. And in the other model it is just application of g and making the resulting word cyclically reduced.

<div class="mw-heading mw-heading3"><h3 id="Connection_to_length_functions">Connection to length functions</h3></div>

Every point in the outer space determines a unique length function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{X}\colon F_{n}\to \mathbb {R} }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l_{X}\colon F_{n}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./46a5ec30e12a91a05367e25e309d437c93d300d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.365ex; height:2.509ex;" alt="{\displaystyle l_{X}\colon F_{n}\to \mathbb {R} }" loading="lazy">. A word in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> determines via the chosen homotopy equivalence a closed path in X. The length of the word is then the minimal length of a path in the free homotopy class of that closed path. Such a length function is constant on each conjugacy class. The assignment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto l_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\mapsto l_{X}}</annotation>

</semantics>

</math><img src="./d15c44d537ab167206ae3fc790d708e84232b7a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.92ex; height:2.509ex;" alt="{\displaystyle X\mapsto l_{X}}" loading="lazy"> defines an embedding of the outer space to some infinite dimensional projective space.

<div class="mw-heading mw-heading3"><h3 id="Simplicial_structure_on_the_outer_space">Simplicial structure on the outer space</h3></div>

In the second model an open simplex is given by all those <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-graphs, which have combinatorically the same underlying graph and the same edges are labeled with the same words (only the length of the edges may differ). The boundary simplices of such a simplex consists of all graphs, that arise from this graph by collapsing an edge. If that edge is a loop it cannot be collapsed without changing the homotopy type of the graph. Hence there is no boundary simplex. So one can think about the outer space as a simplicial complex with some simplices removed. It is easy to verify, that the action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Out} (F_{n})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Out} (F_{n})}</annotation>

</semantics>

</math><img src="./541586b8426c8db8f7ee14198f2ee0791f1585c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:2.843ex;" alt="{\displaystyle \mathrm {Out} (F_{n})}" loading="lazy"> is simplicial and has finite isotropy groups.

<ul><li><a href="Train_track_map" title="Train track map">Train track map</a></li>

<li><a href="Automorphism_group_of_a_free_group" title="Automorphism group of a free group">Automorphism group of a free group</a></li>

<li><a href="Outer_space_(mathematics)" title="Outer space (mathematics)">Outer space</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFLubotzky2011" class="citation cs2">Lubotzky, Alexander (2011-12-15), Dynamics of Aut(Fn) Actions on Group Presentations and Representations, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1109.0155">1109.0155</a></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBestvinaHandel1992" class="citation journal cs1">Bestvina, Mladen; Handel, Michael (1992). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2946562">"Train Tracks and Automorphisms of Free Groups"</a>. Annals of Mathematics. 135 (1): 1–51. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2946562">10.2307/2946562</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2946562">2946562</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFFormanekProcesi1992" class="citation journal cs1">Formanek, Edward; Procesi, Claudio (1992-07-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/0021-8693%2892%2990029-L">"The automorphism group of a free group is not linear"</a>. Journal of Algebra. 149 (2): 494–499. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-8693%2892%2990029-L">10.1016/0021-8693(92)90029-L</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-8693">0021-8693</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFHatcherVogtmann1996" class="citation journal cs1">Hatcher, Allen; Vogtmann, Karen (1996-04-01). <a rel="nofollow" class="external text" href="https://msp.org/pjm/1996/173-2/p09.xhtml">"Isoperimetric inequalities for automorphism groups of free groups"</a>. Pacific Journal of Mathematics. 173 (2): 425–441. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1996.173.425">10.2140/pjm.1996.173.425</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0030-8730">0030-8730</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBestvinaFeighnHandel2000" class="citation journal cs1">Bestvina, Mladen; Feighn, Mark; Handel, Michael (2000). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/121043">"The Tits Alternative for out(Fn) I: Dynamics of Exponentially-Growing Automorphisms"</a>. Annals of Mathematics. 151 (2): 517–623. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9712217">math/9712217</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F121043">10.2307/121043</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/121043">121043</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBridsonVogtmann2000" class="citation journal cs1">Bridson, Martin R.; Vogtmann, Karen (2000-07-15). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0021869300983273">"Automorphisms of Automorphism Groups of Free Groups"</a>. Journal of Algebra. 229 (2): 785–792. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjabr.2000.8327">10.1006/jabr.2000.8327</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-8693">0021-8693</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFBestvinaFeighnHandel2004" class="citation journal cs1">Bestvina, Mladen; Feighn, Mark; Handel, Michael (2004-03-01). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1023/B:GEOM.0000022864.30278.34">"Solvable Subgroups of Out(Fn) are Virtually Abelian"</a>. Geometriae Dedicata. 104 (1): 71–96. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9712219">math/9712219</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AGEOM.0000022864.30278.34">10.1023/B:GEOM.0000022864.30278.34</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9168">1572-9168</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFHatcherVogtmann2004" class="citation journal cs1">Hatcher, Allen; Vogtmann, Karen (2004-12-24). <a rel="nofollow" class="external text" href="https://msp.org/agt/2004/4-2/p25.xhtml">"Homology stability for outer automorphism groups of free groups"</a>. Algebraic & Geometric Topology. 4 (2): 1253–1272. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0406377">math/0406377</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fagt.2004.4.1253">10.2140/agt.2004.4.1253</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1472-2739">1472-2739</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBridsonde_la_Harpe2004" class="citation journal cs1">Bridson, Martin R.; de la Harpe, Pierre (2004-07-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0022123603002167">"Mapping class groups and outer automorphism groups of free groups are C∗-simple"</a>. Journal of Functional Analysis. 212 (1): 195–205. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0306204">math/0306204</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0022-1236%2803%2900216-7">10.1016/S0022-1236(03)00216-7</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-1236">0022-1236</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFFarbHandel2007" class="citation journal cs1">Farb, Benson; Handel, Michael (2007-06-01). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s10240-007-0007-7">"Commensurations of Out(Fn)"</a>. Publications mathématiques. 105 (1): 1–48. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10240-007-0007-7">10.1007/s10240-007-0007-7</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1618-1913">1618-1913</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFKalubaKielakNowak2021" class="citation cs2">Kaluba, Marek; Kielak, Dawid; Nowak, Piotr W. (2021-01-20), On property (T) for $\operatorname{Aut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1812.03456">1812.03456</a></cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFBestvina2023" class="citation web cs1">Bestvina, Mladen (2023-12-15). <a rel="nofollow" class="external text" href="https://ems.press/books/standalone/274/5413">"Groups acting on hyperbolic spaces—a survey"</a>. ems.press. Retrieved 2024-10-13.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFGuirardelHorbez2021" class="citation cs2">Guirardel, Vincent; Horbez, Camille (2021-03-29), <a rel="nofollow" class="external text" href="https://arxiv.org/abs/2103.03696">Measure equivalence rigidity of $\mathrm{Out}(F_N)$</a>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2103.03696">2103.03696</a>, retrieved 2024-12-03</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFCullerVogtmann1986" class="citation journal cs1"><a href="Marc_Culler" title="Marc Culler">Culler, Marc</a>; <a href="Karen_Vogtmann" title="Karen Vogtmann">Vogtmann, Karen</a> (1986). <a rel="nofollow" class="external text" href="http://www.math.cornell.edu/~vogtmann/ScannedPapers/1986.0084.pdf">"Moduli of graphs and automorphisms of free groups"</a> (PDF). <a href="Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a>. 84 (1): 91–119. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986InMat..84...91C">1986InMat..84...91C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01388734">10.1007/BF01388734</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0830040">0830040</a>.</cite></li>

<li><cite id="CITEREFVogtmann2002" class="citation journal cs1">Vogtmann, Karen (2002). <a rel="nofollow" class="external text" href="http://www.math.cornell.edu/~vogtmann/papers/Autosurvey/autosurvey.pdf">"Automorphisms of free groups and outer space"</a> (PDF). <a href="Geometriae_Dedicata" title="Geometriae Dedicata">Geometriae Dedicata</a>. 94: 1–31. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1020973910646">10.1023/A:1020973910646</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1950871">1950871</a>.</cite></li>

<li><cite id="CITEREFVogtmann2008" class="citation cs2">Vogtmann, Karen (2008), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200807/tx080700784p.pdf">"What is … outer space?"</a> (PDF), <a href="Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>, 55 (7): 784–786, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2436509">2436509</a></cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-24" href="https://en.wikipedia.org/wiki/?title=Out(Fn)&oldid=1291890860">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>