## Geometric group theory seminar - 2015/16

### Past talks:

#### September 9: Bogdan Nica (Universität Göttingen), Strong hyperbolicity.

This talk is concerned with the space between CAT(-1) spaces and Gromov hyperbolic spaces. Part of the motivation comes from the analytic theory of hyperbolic groups, and one of the main goals is that of getting hyperbolic groups to act geometrically on hyperbolic spaces with additional CAT(-1) features. Based on joint work with Jan Spakula.

#### September 16: Jingyin Huang (McGill), The pentagon group.

A pentagon group is a group with the following presentation {a,b,c,d,e | [a,b]=[b,c]=[c,d]=[d,e]=[e,a]=1}. It is a right-angled Artin group whose defining graph is a pentagon. There is a nice CAT(0) cube complex X which this group acts properly and cocompact. Our goal is to prove that any other group H which acts properly and cocompactly on X is commensurable to the pentagon group. We will first discuss the motivation for such problems, then sketch a proof of this statement. One key ingredient in the proof is a combination theorem in the spirit of the work of Haglund and Wise. However, due to the lack of (relative) hyperbolicity in our case, we need to adapt their work in a careful way.

#### September 23: Daniel Woodhouse (McGill), A geometric proof of the non-separability of a 3-manifold group.

A group G is residually finite if every non-trivial element survives in a finite quotient. A subgroup H is separable if every element of G that is not in H survives in a finite quotient that eliminates H. I will discuss these notions, focusing on the topological criterion for separability of Scott. A proof that the 3-manifold group of Burns-Karass-Solitar is non-separable will be presented using pictures. To conclude, I will explain how this fits into my work understanding the cubulations of tubular groups, and perhaps talk about the interesting connections with the theory of foliations.

#### September 30: Matthias Nagel (UQAM), Turning 3-manifolds acyclic.

We consider the question whether a given 3-manifold admits a system of local coefficients such that all corresponding homology groups vanish. I will explain the classification of these 3-manifolds and parts of the proof. All necessary background in 3-manifold topology will be given. Based on joint work with Stefan Friedl.

#### October 7, 3:00PM: Jeremy Macdonald (Concordia), Algorithmic problems in nilpotent groups.

Nilpotent groups are one of the most important generalizations of abelian groups. Most algorithmic problems in abelian groups are easy. To what extent does the same hold for nilpotent groups? We will prove that three fundamental problems (word, conjugacy, and membership problems) in nilpotent groups are decidable in logarithmic space and quasilinear time. The majority of the talk, however, will be devoted to background material related to this result, focussing on past and present approaches to computation in nilpotent groups. This is joint work with A. Miasnikov, A. Nikolaev, and S. Vassileva.

#### October 7, 4:15PM: Daniel Kasprowski (Max Planck Institute), On the K-theory of linear groups.

We will show that for every ring R the assembly map in algebraic K-theory is split injective for every subgroup G of a linear group which admits a finite dimensional model for the classifying space for proper actions. For this we will use the concept of finite decomposition complexity, first introduced by Guentner, Tessera, and Yu. It is a coarse invariant of metric spaces and generalizes the notion of finite asymptotic dimension.

#### October 14: Piotr Przytycki (McGill), Cubulations of Artin groups.

I will report on joint work with Kasia Jankiewicz and Jingyin Huang. We exhibit a class of Artin groups which are fundamental groups of nonpositively curved compact cube complexes (shortly npc ccc). We show that 2-dimensional or 3-generator Artin groups outside this class are not fundamental groups of npc ccc, even if we pass to a finite index subgroup. This includes in particular the braid group on 4 strands.

#### October 21: Dooheon Lee (New York University), The Stable Boundaries of CAT(0) Groups.

It is well-known that Gromov-hyperbolic groups have well-defined boundaries. As demonstrated by Croke and Kleiner, however, the same can not be said for CAT(0) groups. To remedy this issue, Charney and Sultan defined a new type of boundary, called the contracting boundary, for CAT(0) groups. Except Gromov-hyperbolic cases, however, the contracting boundaries of CAT(0) groups are not topologically generic. To be precise, suppose G acts on a CAT(0) space X and G is not Gromov-hyperbolic, then the contracting boundary of X is of first Baire category in \partial X. In this paper, we introduce stable boundaries for CAT(0) groups, which coincide with the usual boundary in the cases of Gromov-hyperbolic groups. We provide a sufficient condition for stable boundaries of CAT(0) groups to be topologically generic in the above sense. In particular, we show that the stable boundaries of right angled Artin groups, Coxeter groups, and CAT(0) groups with codimension one free Abelian subgroups are generic when they have rank one axial isometries.

#### October 28: Elisabeth Fink (University of Ottawa), Morse geodesics in lacunary hyperbolic groups.

A geodesic is Morse if quasi-geodesics connecting points on it stay uniformly close. Such geodesics mark hyperbolic directions in the Cayley graph of a group. I will use combinatorial tools to study the geometry of lacunary hyperbolic graded small cancellation groups and show that they contain Morse geodesics. Further I will outline in a simple example an explicit but longer way to find Morse geodesics in such groups. This is joint work with R. Tessera.

#### November 4: Matthew Cordes (Brandeis University), Morse boundaries of geodesic spaces.

I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on the Morse boundary of the mapping class group or briefly describe joint work with David Hume developing the metric Morse boundary.

#### November 11: Svetla Vassileva (Concordia), Log-space conjugacy problem in Grigorchuk groups.

The definition of the Grigorchuk group is deceptively simple: it is generated by several actions on an infinite binary tree. However, it satisfies many uncommon properties: for example it has intermediate growth, but is not finitely presentable. In that light, it is interesting to see that many of the classical computational problems are easy'' in the Grigorchuk group. The word problem is known to be easy'' and we prove the same of the conjugacy problem. Here easy'' means log-space (and hence polynomial-time) decidable. We will introduce the Grigorchuk group and the precise notion of log-space decidability in some detail, following which we will discuss algorithms for the solution of the above-mentioned problems. This is joint work with A. Miasnikov.

#### November 18: Ying Hu (UQAM), Left-orderability and cyclic branched covers over two-bridge knots.

Left-orderability and cyclic branched covers over two-bridge knots A group is called left-orderable if one can put a total order on the set of group elements so that inequalities are preserved by group multiplication on the left. The left-orderability of 3-manifold groups is closely related to the concepts of L-spaces and taut foliations. In this talk, we will show that the n-folded cyclic branched covers of the three sphere over certain two-bridge knots have left-orderble fundamental groups for sufficiently large n.

#### November 25: Mark Powell (UQAM), Metrics on the knot concordance space.

The slice genus of a knot is the minimal genus of a smoothly embedded surface in the 4-ball whose boundary is the knot. This can be used to define an integer valued metric on the space of knots up to concordance. I will describe a refinement of this that uses a 2-complex construction called a grope (roughly, a grope is tower of embedded surfaces) to better approximate a slice disc, leading to a rational valued metric that refines the slice genus metric and reveals non-discrete behaviour.

#### December 2: Michael Brandenbursky (Ben Gurion University), L^p-metrics and autonomous flows.

I will discuss a number of results on the interrelation between the L^p -metric on the group of Hamiltonian diffeomorphisms of surfaces and the subset A of autonomous Hamiltonian diffeomorphisms. In particular, I will show that there are Hamiltonian diffeomorphisms of all surfaces of genus g ≥ 2 or g = 0 lying arbitrarily L^p -far from the subset A, and the diameter of this group equipped with the autonomous metric is infinite. This is a joint work with Egor Shelukhin.

#### January 13: Ilya Kapovich (University of Illinois at Urbana-Champaign), Endomorphisms, train track maps, and fully irreducible monodromies.

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact: an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant \Sigma(G) containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of \Sigma(G) and -\Sigma(G) is connected then for every splitting of G as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible. This talk is based on joint work with Spencer Dowdall and Christopher Leininger.

#### January 20: Tomasz Odrzygózdz (Polish Academy of Science), Why consider the square model for random groups?

We will start with the theory of random groups in general, then introduce several models, including the square model for random groups. This model is a modificiation of the triangular model invented by Zuk. We will show some basic properties of square random groups and then discuss the motivations behind investigating this model (which are related to Property (T), Haggerup Property and residual finiteness). We will compare introduced models and state some conjectures.

#### January 27: Piotr Przytycki (McGill), A dismantlable Rips complex for relatively hyperbolic groups.

We will describe a Rips complex, a thickening of the Cayley graph of a (relatively) hyperbolic group G, with a graph-theoretic property called dismantlability. This guarantees fixed-point properties and implies that the Rips complex is a classifying space for G (with respect to appropriate family). This is joint work with Eduardo Martinez-Pedroza.

#### February 3: Jingyin Huang (McGill), Cubulating extension complex.

One can understand a geometric object by specifying a collection of subspaces, and encoding the intersection pattern of these subspaces in a combinatorial object. Then it is natural to ask whether a morphism of the associated combinatorial object actually comes from a "nice" morphism of the original geometric object. These are called reconstruction problems. A basic example would be the fundamental theorem of projective geometry, where under mild conditions one can reconstruct a semi-linear map from an automorphism of the associated Tits building.
Such reconstruction problems play important role in various quasi-isometry rigidity results. In the case of right-angled Artin groups, such reconstruction results are available by the work of Bestvina, Kleiner and Sageev, as well as Huang, under restrictions on the outer-automorphism of the group. However, there are interesting cases where the reconstruction map fails to exists. In this talk, I will indicate how to use cubulation techniques to get around this issue, and prove a quasi-isometry classification result for more general right-angled Artin groups.

#### February 10: Christopher Smith (McGill), Pairwise Once-Intersecting Arcs on a Sphere

Suppose we are given a punctured sphere with negative Euler characteristic and a distinguished puncture p, and a collection of curves which each have one endpoint on p, and one endpoint on any puncture distinct from p. We demonstrate that, under the condition that the curves pairwise intersect at most once, the maximal size of such a collection of curves is |χ|(|χ|+1), where χ is the Euler characteristic of the punctured sphere. In order to obtain this upper bound, we will look at cyclical orderings of the arcs with endpoints at each puncture q distinct from p, and study how "minimal fish" -- pairs of consecutive arcs which intersect in each such ordering about a puncture q -- affect the number of arcs which can have endpoints at each puncture other than q. This talk is based on joint work with Piotr Przytycki.

#### February 17: Christophe Hohlweg (UQAM), Limit root systems and imaginary cones in Coxeter groups.

Let (W,S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropy cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the limit set of roots E. In this talk we will discuss E and the convex hull of E called the imaginary cone. In particular, we will show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal Coxeter systems. Along the talk, we will discuss open questions and point out the relations of our framework with previously known notions of limit sets associated to W.

#### February 24: Damian Orlef (University of Warsaw), Working with graph spectra - in pursuit of property (T) for new Gromov random groups.

I will review basic properties of graph spectra and their connection to the Kazhdan's property (T) of groups, which is given by the Żuk's spectral criterion for triangular presentations. Then I will focus on the problem of applying this criterion to random groups in naturally defined polygonal models, by means of triangulation. One method of triangulation will be discussed in more detail, showcasing a few techniques useful in estimating the spectral gaps of graphs following certain types of regularities. The main motivation is the search for new densities, at which Gromov random groups have the Kazhdan's property (T). Based on joint work with Piotr Przytycki.

#### March 9: Tomasz Prytuła (Copenhagen University), Coarse product decomposition theorem for systolic complexes and applications.

Let h be a hyperbolic isometry of a systolic complex. In joint work with D. Osajda we show that the minimal displacement set of h decomposes up to quasi-isometry as the product of a tree and the real line. From this we deduce the following two corollaries. For a group G acting properly on a systolic complex we construct a finite dimensional model for the classifying space \underline{\underline{E}}G. If the action is additionally cocompact, we prove that the centralizer of any hyperbolic element is virtually free-by-cyclic, therefore confirming a conjecture of D. Wise. Before outlining these results I shall give some background on classifying spaces for families and systolic complexes.

#### March 16: Kasia Jankiewicz (McGill University), Pseudo-Anosov homeomorphisms.

In this expository talk I will give a definition, some constructions and some properties of pseudo-Anosov homeomorphisms of surfaces.

#### March 23: Jason Behrstock (City University of New York), Asymptotic dimension of mapping class groups.

The goal of this talk will be to describe our recent result proving that the asymptotic dimension of the mapping class group of a closed surface is at most quadratic in the genus (building on and strengthening a prior result of Bestvina-Bromberg giving an exponential estimate). We obtain this result as a special case of a result about the asymptotic dimension of a general class of spaces, which we call hierarchically hyperbolic; this class includes hyperbolic spaces, mapping class groups, Teichmueller spaces endowed with either the Teichmuller or the Weil-Petersson metric, fundamental groups of non-geometric 3-manifolds, RAAGs, etc. We will discuss the general framework and a sketch of how this machinery provides new tools for studying special subclasses, such as mapping class groups. The results discussed are joint work with Mark Hagen and Alessandro Sisto.

#### March 30: Atefeh Mohajeri (McGill), Approximation algorithms for the geodesic problem in free metabelian groups and restricted wreath products.

It has been shown that the geodesic problem for free metabelian groups is NP-hard. Hence, one cannot hope for any polynomial-time algorithm (unless P = NP). We show that there exists a 2-approximation algorithm for the geodesic problem in finitely generated free metabelian groups. We also show that the geodesic problem in the restricted wreath product of a finitely generated non-trivial group A with a finitely generated abelian group B containing free Abelian group of rank 2 is NP-hard. Moreover, we prove that if the geodesic problem is polynomially decidable in A, then there exists a Polynomial Time Approximation Scheme for the geodesic problem in the restricted wreath product of A and B. The results are based on joint work with Olga Kharlampovich.

#### April 6: Richard Webb (University College London), Locally infinite complexes, combinatorial isoperimetric inequalities and CAT(0) metrics.

We shall show that most arc complexes and all but finitely many disc complexes and free splitting complexes do not admit CAT(0) metrics with well-behaved shapes (in particular finitely many shapes). Note that these complexes are Gromov hyperbolic, contractible and any finite group of automorphisms globally fixes a point. We prove the theorem by considering the combinatorial isoperimetric inequality of the complex. On the other hand, the curve complexes and arc-and-curve complexes satisfy a linear combinatorial isoperimetric inequality, and our proof uses the tightening procedure introduced by Masur and Minsky. This suggests a relationship between the combinatorial isoperimetric inequality of a locally infinite complex and the existence of tight geodesics, which is of interest in the study of Out(F_n).

#### April 13: Sang Rae Lee (Texas A&M University), Twisted subgroups of Houghton's groups.

We say a group has type F_n if there exists a K(G,1)-complex having finite n-skeleton. All known examples of groups of type F_{n-1} but not F_n (n>3) contain a subgroup isomorphic to the free abelian group of rank 2. In this talk we consider a twisted subgroup of a Houghton's group to construct a group of type F_3 but not F_4 which does not contain such a subgroup. A Houghton's group H_n consists of translations at infinity'' on n rays of discrete points emanating from the origin on the plane. To show the desired finiteness property we use a CAT(0) cubical complexes associated to Houghton's groups and quasi-isometries between subgroups of Houghton's groups. If time permits we also discuss possible generalization of our construction.

Here is a link to our schedule in 2014/15.
Here is a link to our schedule in 2013/14.