Seminars in Geometric Group Theory
Right-angled Artin groups are perhaps the most ubiquitous manifestations of polyhedral products in geometric group theory and low-dimensional topology. The theory of their subgroups has been of great importance in the last couple of decades. This is especially true with regards to what are known as 'finiteness properties' - meaningful criteria for measuring ways in which infinite groups may behave like finite ones - as well as the theory of three-dimensional manifolds. We will visit some celebrated theorems and, if time allows, discuss problems arising from deck transformations of branched covering maps.
In the 1960s Higman was able to characterize the finitely generated subgroups of finitely presented groups, that is, groups defined using a finite set of generators and finite set of defining relations. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present a result of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the groups of units of inverse monoids defined by presentation where all the defining relators are of the form w=1. I will explain what an inverse monoid is, the motivation for studying this class of inverse monoids, and also outline some of the geometric ideas that we developed in order to prove our results.
Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider ℤ-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.
Given a string Coxeter system (W,S), we construct highly regular quotients of the 1-skeleton of its universal polytope P, which form an infinite family of expander graphs when (W,S) is indefinite and P has finite vertex links. The regularity of the graphs in this family depends on the Coxeter diagram of (W,S). The expansion stems from superapproximation applied to (W,S). This construction is also extended to cover Wythoffian polytopes. As a direct application, we obtain several notable families of expander graphs with high levels of regularity, answering in particular a question posed by Chapman, Linial and Peled positively.
This talk is based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.
This video was produced by the Sydney Mathematical Research Institute, as part of their SMRI seminar series.
Discrete 2-generator subgroups of PSL2(ℝ) have been extensively studied by investigating their action by Möbius transformations on the hyperbolic plane. Due to work of Gilman, Rosenberger, Purzitsky and many others, there is a complete classification of such groups by isomorphism type, and an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℝ) is discrete.
Here we completely classify discrete 2-generator subgroups of PSL2(ℚp) over the p-adic numbers ℚp by studying their action by isometries on the corresponding Bruhat-Tits tree. We give an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℚp) is discrete, and discuss how this can be used to decide whether or not a 2-generator subgroup of SL2(ℚp) is dense.
A locally-finite simple connected graph is said to be k-geodetic for some k ≥ 1, if there are at most k distinct geodesics between any two vertices of the graph. We investigate the properties of hyperbolic groups with k-geodetic Cayley graphs. To begin, we show that k-geodetic graphs cannot have a "ladder-like" geodesic structure with unbounded length. Using this bound, we generalize a well-known result of Papasoglu that states hyperbolic groups with 1-geodetic Cayley graphs are virtually free. We then investigate which elements of the hyperbolic group with k-geodetic Cayley graph commute with a given infinite order element.
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.
The primitivity rank of an element w of a free group F is defined as the minimal rank of a subgroup containing w as an imprimitive element. Recent work of Louder and Wilton has shown that there is a strong connection between this quantity and the subgroup structure of the one-relator group F/≪w≫. In particular, they show that one-relator groups whose defining relation has primitivity rank at least 3 cannot contain Baumslag-Solitar subgroups, leading them to conjecture that such groups are hyperbolic. In this talk, I will show how to confirm and strengthen this conjecture, providing some applications.
In 2014 Doron Puder introduced the notion of primitivity rank π(g) for a non-trivial element g in a free group Fr of rank r.
Namely, π(g) is defined as the smallest rank of a subgroup H of Fr containing g as a non-primitive element, or as ∞ if no such H exists. The set of all subgroups H of Fr as above is denoted Crit(g). It turns out that the primitivity index of an element w ∈ Fr is closely related to the questions about word-hyperbolicity and subgroup properties of the one-relator group < Fr | w=1 >.
We prove that if r≥2 and F2=F(x1, ..., xr) is the free group of rank r, then, as n→∞, for a "random" element wn ∈ Fr of length n with probability tending to 1 one has π(w)=r and Crit(w)={Fr}. We discuss applications of this result to "word measures" on finite symmetric groups SN, defined by such wn.
I'll discuss recent work with Derek Holt that proves that the compressed word problem in groups that are hyperbolic relative to free abelian subgroups can be solved in polynomial time. This result extends results of Lohrey, and of Holt, Lohrey and Schleimer, for free groups and for word hyperbolic groups, and our proof imitates the proofs of those results. I'll define all the terms used in the title, explain background that motivates the result, and outline the methods used in the proof.
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.
A hierarchically hyperbolic structure is some kind of coordinate system on a given metric spaces where the coordinates take values in hyperbolic spaces, and it gives a good understanding of the coarse geometry of the space. I will give a brief introduction to this notion and its consequences, discuss a simple criterion to show that a space or group is hierarchically hyperbolic, and illustrate an application of this criterion to the case of extra-large type Artin groups.
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.
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.
Actions on trees are ubiquitous in group theory. The standard approach to describing them is known as Bass–Serre theory, which presents the group acting on the tree as assembled from its vertex and edge stabilizers. However, a different approach emerges if instead of considering vertex and edge stabilizers as a whole, we focus on local actions, that is, the action of a vertex stabilizer only on the immediate neighbours of that vertex. Groups acting on trees defined by their local actions are especially important as a source of examples of simple totally disconnected locally compact groups, with a history going back to a 1970 paper of Tits. I will go through some highlights of this theory and then present some recent joint work with Simon Smith: we develop a counterpart to Bass–Serre theory for local actions, which describes all possible local action structures of group actions on trees.