Geometric Group Theory

Geometric group theory seminar - 2016/17

The seminar meets each Wednesday at 3 PM in BURN 920

May 10, 2017: Claude Marion (University of Padova), On finite simple images of triangle groups.

Given a triple (a, b, c) of positive integers, a finite group is said to be an (a, b, c)-group if it is a quotient of the triangle group Ta,b,c = hx, y, z : x a = y b = z c = xyz = 1i. Let G0 = G(p r ) be a finite quasisimple group of Lie type with corresponding simple algebraic group G. Given a positive integer a, let G[a] = {g ∈ G : g a = 1} be the subvariety of G consisting of elements of order dividing a, and set ja(G) = dim G[a] . Given a triple (a, b, c) of positive integers, we conjectured a few years ago that if ja(G) +jb(G) +jc(G) = 2 dim G then given a prime p there are only finitely many positive integers r such that G(p r ) is an (a, b, c)-group. We present some recent progress on this conjecture and related results: in particular the conjecture holds for finite simple groups.

April 5, 2017: Jonah Gaster (Boston College), New bounds for `homotopical Ramsey theory' on surfaces.

Farb and Leininger asked: How many distinct (isotopy classes of) simple closed curves on a finite-type surface S may pairwise intersect at most k times? Przytycki has shown that this number grows at most as a polynomial in |\chi(S)| of degree k^{2}+k+1. We present narrowed bounds by showing that the above quantity grows slower than |\chi(S)|^{3k}. The most interesting case is that of k=1, in which case the size of a `maximal 1-system' grows sub-cubically in |\chi(S)|. Following Przytycki, the proof uses the hyperbolic geometry of a surface of negative Euler characteristic essentially. In particular, we require bounds for the maximum size of a collection of curves of length at most L on a hyperbolic surface homeomorphic to S of the form F(L) \cdot |\chi|, a point of view that yields intriguing questions in its own right. This is joint work with Tarik Aougab and Ian Biringer.

March 29, 2017: Damian Orlef (Polish Academy of Sciences), Random groups, cyclic orders and actions on the circle.

We pursue the following question: can random groups in the Gromov density model act non-trivially on the circle by homeomorphisms? If one considers just faithful actions by orientation-preserving homeomorphisms, then it is the same as asking about existence of cyclic total left-orders on random groups. In perhaps the first step towards answering this question, we show that random triangular groups are not cyclically left-ordered for densities in range (1/3,4/9). We also discuss the case of smooth actions. Based on joint work with Piotr Przytycki. Work in progress.

March 22, 2017: Olga Kharlampovich (Grad Center and Hunter College CUNY), Equations in Group Algebras.

We show that the Diophantine problem (decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc. We will also show that geometry of a limit group G is definable in the language of the group ring K(G) for an infinite field K. These are joint results with A. Miasnikov.

March 15, 2017: Kasra Rafi (University of Toronto), Geometry of the Thurston metric on Teichmüller space.

Teichmüller space can be equipped with a metric using the hyperbolic structure of a Riemann surface, as opposed to the conformal structure that is used to define the Teichmüller metric. This metric, which is asymmetric, was introduced by Thurston and has not been studied as extensively as Teichmüller metric or the Weil-Petersson metric. However, it equips Teichmüller space with a distinctive and rich structure. We give a survey of some recent results and discuss some open problems and conjectures.

March 8, 2017: Phillip Wesolek (Binghamton University), The structure of simple totally disconnected locally compact groups via embeddings with dense image.

(Joint work with P.-E. Caprace and C. Reid) The collection of topologically simple totally disconnected locally compact (t.d.l.c.) groups which are compactly generated and non-discrete, denoted by \mathscr{S}, forms a rich and compelling class of locally compact groups. Members of this class include the simple algebraic groups over non-archimedean local fields, the tree almost automorphism groups, and groups acting on CAT(0) cube complexes. In recent years, a general theory for these groups, which considers the interaction between the geometry and the topology, has emerged. In this talk, we study this interaction by considering the non-discrete t.d.l.c. groups H which admit a continuous embedding with dense image into some group G \in \mathscr{S}. We indeed consider a class \mathscr{R} which contains all such t.d.l.c. groups and show \mathscr{R} enjoys many of the same properties previously established for \mathscr{S}. Using these more general results, new restrictions on the members of \mathscr{S} are obtained. For any G\in \mathscr{S}, we prove that any infinite Sylow pro-p subgroup of a compact open subgroup of G is not solvable. We prove further that there is a finite set of primes \pi such that every compact subgroup of \mathscr{Aut}(G) is virtually pro-\pi.

March 1, 2017: NO SEMINAR - reading week

February 22, 2017: Assaf Bar-Natan (McGill University), Arcs on punctured disks pairwise intersecting twice

In this talk, we will discuss systems of essential simple arcs on a n-punctured disk pairwise intersecting at most twice. These systems will naturally lead us to a structure theorem similar to Greendlinger's Lemma about disk diagrams whose hyperplanes are simple arcs intersecting at most twice. This is joint work with Piotr Przytycki.

February 15, 2017: Piotr Przytycki (McGill University), Tame automorphism group.

We study the group of polynomial automorphisms of C^3 generated by affine maps and all (x,y,z)->(x+P(y,z),y,z). We exhibit a contractible hyperbolic 2-complex on which that group acs. We also find a loxodromic weakly properly discontinuous element. This is joint work with Stephane Lamy.

February 8, 2017: Kasia Jankiewicz (McGill University), Graph coloring problem and fibering right angled Coxeter groups.

I will describe a simple graph coloring problem. Its solution, if exists, provides a virtual algebraic fibration of the right angled Coxeter group associated to the graph. This is a joint work with Sergey Norin and Daniel Wise.

February 1, 2017: Anthony Conway (University of Geneva), The Burau representation and twisted Alexander polynomials.

After reviewing the definition of the braid group, I will recall a topological construction of the Burau representation and its relation to the Alexander polynomial. If time permits, I will then discuss a generalisation of this result involving twisted Alexander polynomials.

January 25, 2017: Forte Shinko (McGill University), Borel complexity of boundary actions of hyperbolic groups.

An active area of descriptive set theory seeks to compare and classify the complexity of Borel equivalence relations via the notion of Borel reduction. Given a group acting geometrically on a hyperbolic space, we will investigate the Borel complexity of the induced action on the Gromov boundary. This is joint work with Jingyin Huang and Marcin Sabok.

January 18, 2017: Nima Hoda (McGill University), Quadric Complexes.

Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional CAT(0) cube complexes and may be viewed as the square analog of systolic complexes. I will define and present some properties of quadric complexes and of groups which act on them. Along the way I will discuss or mention b-dismantlability of the 1-skeleton, a classifying space for the finite subgroups and that C4-T4 small cancellation groups are quadric.

January 11, 2017: Daniel Wise (McGill University), Virtual Limit Groups. Part II.

I will describe a family of groups that have a finite index subgroup which is a limit group. The family includes every word-hyperbolic group that splits as a free product of two free groups amalgamating a cyclic subgroup.

November 30, 2016: Daniel Wise (McGill University), Virtual Limit Groups.

November 23, 2016 at 4 PM: Sylwia Antoniuk (Adam Mickiewicz University), From the random 3-uniform hypergraph to the random triangular group.

We consider the triangular model of a random group and study the threshold for the property of not being free. We show that in some sense this threshold coincides with the threshold for the appearance of a 2-core in the random 3-uniform hypergraph. This is a joint work with Tomasz Łuczak, Tomasz Prytuła, Piotr Przytycki, and Bartek Zaleski.

November 23, 2016: Damian Orlef (Polish Academy of Sciences), Kazhdan's property (T) of random groups in the square model for d>5/12.

A random group in the square model is obtained by fixing a set of n generators and introducing at random about (2n)^(4d) relations of length 4 between them, where d is a fixed parameter called the density and n tends to infinity. By results of T. Odrzygóźdź, if d<1/2, then these groups are with overwhelming probability (w.o.p.) infinite and hyperbolic. We prove that for d>5/12 the random groups G in the square model have w.o.p. Kazhdan's property (T). The proof proceeds by constructing a triangular group H, which maps onto finite index subgroup of G and verifying that the Żuk's spectral criterion can be successfully applied to yield Kazhdan's property of H. The verification proceeds by analyzing random walks on the link of H, in the spirit of Broder and Shamir. Joint work with T. Odrzygóźdź and P. Przytycki.

November 21, 2016 (Monday at 4:30 in 1234): Pierre-Emmanuel Caprace (Université catholique de Louvain), Commensuration index growth.

The title refers to an asymptotic invariant of a pair (G, U) consisting of a group G and a commensurated subgroup U < G such that G is generated by finitely many cosets of U. The goal of the talk is to introduce this invariant, discuss examples and present the first results of its investigation. This is based on joint work with Colin Reid and Phillip Wesolek.

November 21, 2016 (Monday at 3:30 in 1234): Nicolas Radu (Université catholique de Louvain), A locally non-Desarguesian A2-tilde building admitting a uniform lattice.

An A2-tilde building is a simply connected simplicial complex of dimension 2 such that each sphere of radius 1 centered at a vertex is isomorphic to the incidence graph of a projective plane. In 1986, William Kantor asked the problem of constructing an A2-tilde building with a cocompact lattice and whose local projective planes are finite and non-Desarguesian. In this talk, I will tell the exciting story of how such an A2-tilde building could be discovered.

November 16, 2016: NO SEMINAR (this instead)

November 10, 2016 (Thursday 4:15 PM, room 1234): Daniel Woodhouse (Technion), The reflection trick for non-positively curved cube complexes.

I will show how to embed a compact npc cube complex into a npc curved cube complex homeomorphic to a closed manifold by local isometry.

November 9, 2016: PhD Oral Defence of Daniel Woodhouse.

November 2, 2016: Tomasz Odrzygóźdź (Polish Academy of Sciences, Sharp threshold for Property (T) in the hexagonal model for random groups.

A random group in the hexagonal model is a group given by the presentation where R is a set of random words of length 6 over the set S. We consider properties of such a group as the cardinality of S goes to infinty. Our main goal in the presentation is to prove that as the cardinality of the set R increases (we increase the density defined as d =(1\6) *log_{|S|}(|R|)) there is a sharp transition between not having Property (T) and having Property (T) for such group (this threshold is at density d=1/3). First we will present a quick survey about what is known at the moment about Property (T) for random groups. To proof thie main result we will present a new method of constructing some good system of walls on the Cayley complex of a random group. This will allows us to find a proper action of a random group on a CAT(0) cube complex. The main idea behind constructing our system of walls is to take hypergraphs in the Cayley complex (as Wise and OIllivier did in their paper about cubulating random groups) and then correct them to make them embedded trees.

October 26, 2016: Rita Gitik (University of Michigan), On intersections of conjugate subgroups.

We define a new invariant of a conjugacy class of subgroups which we call the weak width and prove that a quasiconvex subgroup of a negatively curved group has finite weak width in the ambient group. Utilizing the coset graph and the geodesic core of a subgroup we give an explicit algorithm for constructing a finite generating set for an intersection of a quasiconvex sub-group of a negatively curved group with a conjugate. Using that algorithm we construct algorithms for computing the weak width, the width and the height of a quasiconvex subgroup of a negatively curved group. These algorithms decide if a quasiconvex subgroup of a negatively curved group is almost malnormal in the ambient group.

October 19, 2016: Eduardo Martinez-Pedroza (Memorial University of Newfoundland), On Hartnell's Firefighting Problem.

The firefighter game on graphs was introduced by Bert Hartnell in 1995. Briefly, an fire breaks out at a finite set of vertices; at each time interval n ≥ 1, a fixed number of vertices p which are not on fire become protected; then the fire spreads to all unprotected neighbors of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The player looks for an strategy to contain the fire. We study this game and some of its variations and show that the existence of a number p such that there is a winning strategy for any initial fire is a quasi-isometry invariant in the class of infinite graphs of bounded degree. Then we start exploring these games from geometric group theory point of view. The talk includes joint work with Danny Dyer and Brandon Thorne, and joint work with Tomasz Prytula.

October 12, 2016: Liam Watson (Université de Sherbrooke), Heegaard Floer homology and graph manifolds.

Between you and me, I always wanted to be a geometric group theorist. On that note, there is a conjectural relationship between Heegaard Floer homology and left-orderable three-manifold groups, which is now known to hold for all graph manifolds. That is, a closed orientable graph manifold Y is an L-space if and only if the fundamental group of Y is not-left-orderable. The latter condition can be hard to check in general, but in this setting the inequality dimHF(Y)>|H_1(Y)| provides a certificate that the group pi_1(Y) is left-orderable. I'll explain what these symbols mean in an attempt to sell the machinery to geometric group theorists.

October 5, 2016: Erika Kuno (Tokyo Institute of Technology), Uniform hyperbolicity for curve graphs of non-orientable surfaces.

We will talk about Gromov hyperbolicity for curve graphs of non-orientable surfaces. Hensel-Przytycki-Webb in 2015 proved that the curve graphs of orientable surfaces are 17-hyperbolic. We apply their arguments to non-orientable surfaces, and prove that the curve graphs of non-orientable surfaces are also 17-hyperbolic. In this talk, we will focus on the difference from the orientable surface case.

September 28, 2016: Bogdan Nica (McGill University), Rapid Decay, Rapidly

I will give a sketchy overview of the analytic property of Rapid Decay. The case of free groups will be discussed in some detail.

September 21, 2016 at 4 PM: Jarosław Buczyński (University of Warsaw and Polish Academy of Sciences), Constructions of k-regular maps using algebraic geometry.

A continuous map R^m -> R^N or C^m -> C^N is called k-regular if the images of any k distinct points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m. During the talk, using the methods of algebraic geometry, we will construct k-regular maps. We will relate the upper bounds on N with the dimension of the parametrising spaces of certain finite C-algebras. The computation of the dimension of this space is explicit for k< 10, and we provide explicit examples for k at most 5. We will also provide upper bounds for arbitrary m and k.

September 21, 2016 at 3 PM: Cristobal Rivas (Universidad de Santiago de Chile), Orderings and rigidity of one dimensional group actions.

Let G be a countable group. It is known that G admits a faithful action by homeomorphisms on the circle (resp. on the line), if and only if it enjoys a (total) circular ordering (resp. left-ordering). In this talk we will explore the relationship between the space of actions of G on the line or the circle and the corresponding space of orderings. We will give a characterization in dynamical terms of the fact that G admits an isolated orderings. Essentially, an orderings is isolated if and only if its corresponding action enjoys a strong form of rigidity. We will deduce some known as well as some new results from it.

September 14, 2016: Damian Osajda (Wroclaw University), Embedding some infinite graphs into finitely generated groups. Part II

I will present a construction of finitely generated groups with Cayley graphs containing some given infinite sequences of finite connected graphs. The motivation for such construction is looking for groups with various exotic properties, e.g. groups containing expanders. The construction uses a graphical small cancellation theory.