Udo Baumgartner

Contraction groups and scales of automorphisms of totally disconnected locally compact groups

We study contraction groups for automorphisms of totally disconnected locally compact groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.

Contraction groups in complete Kac-Moody groups

Let G be an abstract Kac-Moody group over a finite field and x G the closure of the image of G in the automorphism group of its positive building. We show that if the Dynkin diagram associated to G is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in x G which are not topologically periodic are not closed. (In such groups there always exist elements that are not topologically periodic.) Mathematics Subject Classification (2000). 22D05, 22D45, 20E36.

Hecke Algebras of Group Extensions

Abstract We describe the Hecke algebra ℋ(Γ,Γ0) of a Hecke pair (Γ,Γ0) in terms of the Hecke pair (N,Γ0) where N is a normal subgroup of Γ containing Γ0. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ⊂ Γ/N satisfies S −1 S = Γ/N, we show that ℋ (Γ,Γ0) is the twisted crossed product of ℋ (N,Γ0) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.

Geometric characterization of flat groups of automorphisms

If ℋ is a flat group of automorphisms of finite rank n of a totally disconnected, locally compact group G, then each orbit of ℋ in the metric space If ℋ is a flat group of automorphisms of finite rank n of a totally disconnected, locally compact group G, then each orbit of ℋ in the metric space ℬ(G) of compact, open subgroups of G is quasi-isometric to n-dimensional Euclidean space. In this note we prove the following partial converse: Assume that G is a totally disconnected, locally compact group such that ℬ (G) is a proper metric space and let ℋ be a group of automorphisms of G such that some (equivalently every) orbit of ℋ in ℬ(G) is quasi-isometric ton-dimensional Euclidean space, then ℋ has a finite index subgroup which is flat of rank n. We can draw this conclusion under weaker assumptions. We also single out a naturally defined flat subgroup of such groups of automorphisms.(G) of compact, open subgroups of G is quasi-isometric to n-dimensional Euclidean space. In this note we prove the following partial converse: Assume that G is a totally disconnected, locally compact group such that ℬ(G) is a proper metric space and let ℋ be a group of automorphisms of G such that some (equivalently every) orbit of ℋ in ℬ(G) is quasi-isometric ton-dimensional Euclidean space, then ℋ has a finite index subgroup which is flat of rank n. We can draw this conclusion under weaker assumptions. We also single out a naturally defined flat subgroup of such groups of automorphisms.

A compactly generated group whose Hecke algebras admit no bounds on their representations

We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a C*-enveloping algebra.

Totally Disconnected, Locally Compact Groups as Geometric Objects

This survey outlines a geometric approach to the structure theory of totally disconnected, locally compact groups. The content of my talk at Geneva is contained in Section 3.

Scale-multiplicative semigroups and geometry: automorphism groups of trees

A scale-multiplicative semigroup in a totally disconnected, locally compact group

The direction of an automorphism of a totally disconnected locally compact group

G

Flat rank of automorphism groups of buildings

is one for which the restriction of the scale function on

Hyperbolic groups have flat-rank at most 1

G