Hecke algebras from groups acting on trees and HNN extensions (original) (raw)

2008

We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies past algebraic and analytic approaches by focusing on the common geometric thread.The results have implications for the general theory of totally disconnected locally compact groups.

TWISTED POINCARE SERIES AND ZETA FUNCTIONS ON FINITE QUOTIENTS OF BUILDINGS

Journal of Algebraic Combinatorics, 2018

In the case where G =SL 2 (F) for a non-archimedean local field F and Γ is a discrete torsion-free cocompact subgroup of G, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of G by the action of Γ, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of G. We show how this can be generalized to other split simple algebraic groups of rank two over F , and formulate a conjecture about how this might be generalized to groups of higher rank.

C ∗ -Simple Groups: Amalgamated Free Products, HNN Extensions, and Fundamental Groups of 3-MANIFOLDS

2009

We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions. Much of our analysis deals with conditions on an action of a group Gamma\GammaGamma on a tree TTT which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree TTT and on its boundary partialT\partial TpartialT, and faithfulness in a strong sense. An important step in this analysis is to identify automorphism of TTT which are \emph{slender}, namely such that their fixed-point sets in partialT\partial TpartialT are nowhere dense for the shadow topology.

On parabolic subgroups and Hecke algebras of some fractal groups

Arxiv preprint math/9911206, 1999

Abstract: We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in ...

Se p 20 02 From Fractal Groups to Fractal Sets

2008

3 Self-similar sets and (semi)group actions 8 3. 11 Finitely presented dynamical systems and semi-Markovian spaces 57 12 Spectra of Schreier graphs and Hecke type operators 59 12. The idea of self-similarity is one of the most fundamental in the modern mathematics. The notion of " renormalization group " , which plays an essential role in quantum field theory, statistical physics and dynamical systems, is related to it. The notions of fractal and multi-fractal, playing an important role in singular geometry, measure theory and holomorphic dynamics, are also related. Self-similarity also appears in the theory of C *-algebras (for example in the representation theory of the Cuntz algebras) and in many other branches of mathematics. Starting from 1980 the idea of self-similarity entered algebra and began to exert great influence on asymptotic and geometric group theory. The aim of this paper is to present a survey of ideas, notions and results that are connected to self-simil...

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