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In mathematics, a totally disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff.

Interest centres on locally compact totally disconnected groups (variously referred to as groups of td-type,cite-ref-1[1] locally profinite groups,cite-ref-bushnellhenniart-2-0[2] or t.d. groupscite-ref-3[3]). The compact case has been heavily studied – these are the profinite groups – but for a long time not much was known about the general case. A theorem of van Dantzigcite-ref-dantzig-4-0[4] from the 1930s, stating that every such group contains a compact open subgroup, was all that was known. Then groundbreaking work by George Willis in 1994,cite-ref-5[5] opened up the field by showing that every locally compact totally disconnected group contains a so-called tidy subgroup and a special function on its automorphisms, the scale function, giving a quantifiable parameter for the local structure. Advances on the global structure of totally disconnected groups were obtained in 2011 by Caprace and Monod, with notably a classification of characteristically simple groups and of Noetherian groups.cite-ref-6[6]

• Locally compact case

• Tidy subgroups

• The scale function

• Calculations and applications

• Notes

• References

In a locally compact, totally disconnected group, every neighbourhood of the identity contains a compact open subgroup. Conversely, if a group is such that the identity has a neighbourhood basis consisting of compact open subgroups, then it is locally compact and totally disconnected.cite-ref-bushnellhenniart-2-1[2]

Let G be a locally compact, totally disconnected group, U a compact open subgroup of G and α α {\displaystyle \alpha } a continuous automorphism of G.

Define:

U + = ⋂ ⋂ n ≥ ≥ 0 α α n ( U ) {\displaystyle U_{+}=\bigcap _{n\geq 0}\alpha ^{n}(U)}

U − − = ⋂ ⋂ n ≥ ≥ 0 α α − − n ( U ) {\displaystyle U_{-}=\bigcap _{n\geq 0}\alpha ^{-n}(U)}

U + + = ⋃ ⋃ n ≥ ≥ 0 α α n ( U + ) {\displaystyle U_{++}=\bigcup _{n\geq 0}\alpha ^{n}(U_{+})}

U − − − − = ⋃ ⋃ n ≥ ≥ 0 α α − − n ( U − − ) {\displaystyle U_{--}=\bigcup _{n\geq 0}\alpha ^{-n}(U_{-})}

U is said to be tidy for α α {\displaystyle \alpha } if and only if U = U + U − − = U − − U + {\displaystyle U=U_{+}U_{-}=U_{-}U_{+}} and U + + {\displaystyle U_{++}} and U − − − − {\displaystyle U_{--}} are closed.

The index of α α ( U + ) {\displaystyle \alpha (U_{+})} in U + {\displaystyle U_{+}} is shown to be finite and independent of the U which is tidy for α α {\displaystyle \alpha } . Define the scale function s ( α α ) {\displaystyle s(\alpha )} as this index. Restriction to inner automorphisms gives a function on G with interesting properties. These are in particular:

Define the function s {\displaystyle s} on G by s ( x ) := s ( α α x ) {\displaystyle s(x):=s(\alpha _{x})} , where α α x {\displaystyle \alpha _{x}} is the inner automorphism of x {\displaystyle x} on G.

• s {\displaystyle s} is continuous.

• s ( x ) = 1 {\displaystyle s(x)=1} , whenever x in G is a compact element.

• s ( x n ) = s ( x ) n {\displaystyle s(x^{n})=s(x)^{n}} for every non-negative integer n {\displaystyle n} .

• The modular function on G is given by Δ Δ ( x ) = s ( x ) s ( x − − 1 ) − − 1 {\displaystyle \Delta (x)=s(x)s(x^{-1})^{-1}} .

The scale function was used to prove a conjecture by Hofmann and Mukherja and has been explicitly calculated for p-adic Lie groups and linear groups over local skew fields by Helge Glöckner.

cite-note-11. ↑ Cartier 1979, §1.1

cite-note-bushnellhenniart-22. ↑ Bushnell & Henniart 2006, §1.1

cite-note-33. ↑ Borel & Wallach 2000, Chapter X

cite-note-dantzig-44. ↑ van Dantzig 1936, p. 411

cite-note-55. ↑ Willis 1994

cite-note-66. ↑ Caprace & Monod 2011

• citerefvan-dantzig1936van Dantzig, David (1936), "Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen", Compositio Mathematica, 3: 408–426

• citerefborelwallach2000Borel, Armand; Wallach, Nolan (2000), Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical surveys and monographs, vol. 67 (Second ed.), Providence, Rhode Island: American Mathematical Society, ISBN 978-0-8218-0851-1, MR 1721403

• citerefbushnellhenniart2006Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120

• citerefcapracemonod2011Caprace, Pierre-Emmanuel; Monod, Nicolas (2011), "Decomposing locally compact groups into simple pieces", Mathematical Proceedings of the Cambridge Philosophical Society, 150 (1): 97–128, arXiv:0811.4101, Bibcode:2011MPCPS.150...97C, doi:10.1017/S0305004110000368, MR 2739075

• citerefcartier1979Cartier, Pierre (1979), "Representations of p {\displaystyle {\mathfrak {p}}} -adic groups: a survey", in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions (PDF), Proceedings of Symposia in Pure Mathematics, vol. 33, Part 1, Providence, Rhode Island: American Mathematical Society, pp. 111–155, ISBN 978-0-8218-1435-2, MR 0546593

• citerefwillis1994Willis, G. (1994), "The structure of totally disconnected, locally compact groups", Mathematische Annalen, 300: 341–363, doi:10.1007/BF01450491