Totally disconnected group

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,[1] locally profinite groups,[2] or t.d. groups[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 Dantzig[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, 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.

Locally compact case

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.[2]

Tidy subgroups

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

Define:

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

U is said to be tidy for 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 scale function

The index of α ( U + ) {\displaystyle \alpha (U_{+})} in U + {\displaystyle U_{+}} is shown to be finite and independent of the U which is tidy for α {\displaystyle \alpha } . Define the scale function s ( 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 x ) {\displaystyle s(x):=s(\alpha _{x})} , where α x {\displaystyle \alpha _{x}} is the inner automorphism of x {\displaystyle x} on G.

Properties

  • 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 ( x ) = s ( x ) s ( x − 1 ) − 1 {\displaystyle \Delta (x)=s(x)s(x^{-1})^{-1}} .

Calculations and applications

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.

Notes

References

  • van Dantzig, David (1936), "Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen", Compositio Mathematica, 3: 408–426
  • Borel, 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
  • Bushnell, 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
  • Caprace, Pierre-Emmanuel; Monod, Nicolas (2011), "Decomposing locally compact groups into simple pieces", Math. Proc. Cambridge Philos. Soc., 150: 97–128, arXiv:0811.4101, Bibcode:2011MPCPS.150...97C, doi:10.1017/S0305004110000368, MR 2739075
  • Cartier, 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
  • G.A. Willis - The structure of totally disconnected, locally compact groups, Mathematische Annalen 300, 341-363 (1994)