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.
Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required.
The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the 'uniform discreteness' of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham.
We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism F and use it to characterize when the homological stretch factor of F is equal to its geometric stretch factor.
The boundary of a Gromov hyperbolic group carries a canonical family of metrics which determine the quasi-isometry type of the group. Pansu's conformal dimension of the boundary gives a natural and important quasi-isometric invariant. I will discuss how this invariant behaves when the group splits over two-ended subgroups (i.e. when the boundary has local cut points), and applications to the question of Bonk and Kleiner asking for a characterization of when this dimension equals one.
n this talk we will discuss:
1. Tensor decompositions of locally matrix algebras and their parametrization by Steinitz numbers.
2. Automorphisms and derivations of locally matrix algebras.
3. Automorphisms and derivations of Mackey algebras and Mackey groups. In particular, we describe automorphisms of all infinite simple finitary torsion groups (in the classification of J.Hall) and derivations of all infinite-dimensional simple finitary Lie algebras (in the classification of A.Baranov and H.Strade).
Baumslag-Solitar groups BS(p,q) =< a,t | tapt-1 = aq > were first introduced as examples of non-Hopfian groups. They may be described using graphs of cyclic groups. In analogy with the study of Out(Fn) one can study their automorphisms through their action on an "outer space". After introducing generalized Baumslag-Solitar groups and their actions on trees, I will present an analogue of a Whitehead algorithm which takes an element of a free group and decides whether there exists a free factor which contains that element.
The contraction subgroup for x in the locally compact group, G,
con(x)={ g ∈ G ∣ xngx−n → 1 as n → ∞ },
and the Levi subgroup is
lev(x)={ g ∈ G ∣ {xngx−n}n∈ℤ has compact closure }.
The following will be shown. Let G be a totally disconnected, locally compact group and x ∈ G. Let y ∈ lev(x). Then there are x′ ∈ G and a compact subgroup, K ≤ G such that:
-K is normalized by x′ and y,
-con(x′)=con(x) and lev(x′)=lev(x) and
-the group ⟨x′,y,K⟩ is abelian modulo K, and hence flat.
If no compact open subgroup of G normalized by x and no compact open subgroup of lev(x) normalized by y, then the flat-rank of ⟨x′,y,K⟩ is equal to 2.
In 1993 Brink and Howlett proved that finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognizing the language of reduced words in the Coxeter group. This automaton is constructed in terms of the remarkable set of "elementary roots" in the associated root system.
In this talk we outline the construction of Brink and Howlett. We also describe the minimal automaton recognizing the language of reduced words, and prove necessary and sufficient conditions for the Brink-Howlett automaton to coincide with this minimal automaton. This resolves a conjecture of Hohlweg, Nadeau, and Williams, and is joint work with Yeeka Yau.
In the 90s, Nebbia conjectured that a group of tree automorphisms acting transitively on the tree's boundary must be of type I, that is, its unitary representations can in principal be classified. For key examples, such as Burger-Mozes groups, this conjecture is verified. Aiming for a better understanding of Nebbia's conjecture and a better understanding of representation theory of groups acting on trees, it is natural to ask whether there is a characterization of type I groups acting on trees. In 2016, we introduced in collaboration with Cyril Houdayer a refinement of Nebbia's conjecture to a trichotomy, opposing type I groups with groups whose von Neumann algebra is non-amenable. For large classes of groups, including Burger-Mozes groups, we could verify this trichotomy.
In this talk, I will motivate and introduce the conjecture trichotomy for groups acting on tress and explain how von Neumann algebraic techniques enter the picture.
The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SLn(ℤ), for n ≥ 3, SLn(ℤ[1/p]) for n ≥ 2, their finite-index subgroups and many more. A number of remarkable results about them have been proven including: Weil local rigidity, Mostow strong rigidity, Margulis superrigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity. We will add a new type of rigidity: 'first order rigidity'. Namely if D is such a non-uniform characteristic zero arithmetic group and L a finitely generated group which is elementary equivalent to D then L is isomorphic to D. This stands in contrast with Zlil Sela's remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non-isomorphic finitely generated groups which are elementarily equivalent to them.