Tag - Infinite groups
Due to work of Gilman, Rosenberger, Purzitsky and many others, discrete two-generator subgroups of PSL2(ℝ) have been completely classified by studying their action by Möbius transformations on the hyperbolic plane. Here we aim to classify discrete two-generator subgroups of PSL2(ℚp) by studying their action by isometries on the Bruhat-Tits tree. We first give a general structure theorem for two-generator groups acting by isometries on a tree, which relies on certain Klein-Maskit combination theorems. We will then discuss how this theorem can be applied to determine discreteness of a two-generator subgroup of PSL2(ℚp).
Let C be a smooth, projective, and geometrically connected curve defined over a finite field F. For each closed point P∞ of C, let R be the ring of functions that are regular outside P∞, and let K be the completion path P∞ of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)∖T, where T is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)∖T is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property. In this exposition we describe the quotient graph H∖T defined from the action on T of the group H consisting of matrices that are upper triangular modulo I, where I is an ideal of R. More specifically, we give an explicit formula for the cusp number H∖T. Then By, using Bass-Serre theory, we describe the combinatorial structure of H. These groups play, in the function field context, the same role as the Hecke Congruence subgroups of SL2(ℤ). Moreover, not that the groups studied by Serre correspond to the case where the ideal I coincides with the ring R.
This is joint work with Gili Golan-Polak. We describe the so-called closed subgroups of F. In particular, we construct a subgroup of F with easily decidable membership problem and undecidable conjugacy problem.
Subgroups of Thompson's group F can be very complex. We give a family of elementary amenable subgroups that models a large initial segment of countable ordinals. The family models not only the order structure but also the basic operations of sum, product and exponentiation with base ω. Part of the appeal of the family is its ease of description.
In joint work with Brendan Mallery, we introduce and study so called "shift-similar" groups. Self-similar groups are a well known class of groups, which in particular interact nicely with Higman-Thompson groups, and we introduce shift-similar groups as an analog that interacts nicely with Houghton groups. Shift-similar groups actually turn out to have many properties that self-similar groups do not, for example every finitely generated group embeds into some finitely generated shift-similar group, and there exist uncountably many finitely generated shift-similar groups. In this talk I will recall some background on self-similar groups, introduce shift-similar groups and the Houghton-like groups they produce, and discuss the aforementioned results plus some results about amenability. I will also highlight some open questions.
In the 1980s Madlener and Otto asked for an algebraic characterisation of groups presented by finite, convergent, length-reducing rewriting systems, conjecturing that they are exactly the plain groups (free product of finitely many finite groups and infinite cyclic groups).
I will describe some recent results with Adam Piggott (ANU) on new geometric, algebraic and algorithmic properties of groups presented by (inverse-closed) finite, convergent, length-reducing rewriting systems.
I present a method that allows to prove that some groups are residually nilpotent. In particular, it can be applied to show that the free ℚ-groups are residually (torsion-free nilpotent). This solves a conjecture of Baumslag.
We prove that the Diophantine problem for quadratic equations in unimodular and metabelian Baumslag-Solitar groups BS(m,n) is decidable and belongs to NP. Furthermore, the problem is polynomial-time decidable if |m|=|n|=1 and is NP-hard otherwise.
Let G be the fundamental group of a closed orientable surface of genus at least 2, and α an automorphism of G. In a celebrated result, Thurston showed that the mapping torus G⋊αℤ is hyperbolic if and only if no power of α preserves a non-trivial conjugacy class. In this talk, I will describe joint work with François Dahmani, where we show that if G is torsion-free hyperbolic, then G⋊αℤ is relatively hyperbolic with optimal parabolic subgroups.
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