McGill Geometric Group Theory Seminar
The seminar takes place on Wednesday at 3 PM in 1104 Burnside Hall, unless otherwise noted below.
Seminar organizers: Macarena Covadonga Robles Arenas, Sami Douba, Daniel Wise
Seminar organizers: Macarena Covadonga Robles Arenas, Sami Douba, Daniel Wise
Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A common model geometry for groups $G$ and $G'$ is a proper geodesic metric space on which $G$ and $G'$ act geometrically. A group $G$ is action rigid if any group $G'$ that has a common model geometry with $G$ is abstractly commensurable to $G$. We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space. This is joint work with Emily Stark.
One of the mainstream and modern tools in the study of non-abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots.
Let $S$ be a compact oriented surface. In this talk, I will discuss several invariant metrics and quasi-morphisms on the identity component $\mathrm{Diff}_0(S, \mathrm{area})$ of the group of area-preserving diffeomorphisms of $S$. In particular, I will show that some quasi-morphisms on $\mathrm{Diff}_0(S, \mathrm{area})$ are related to the topological entropy. More precisely, I will discuss a construction of infinitely many linearly independent quasi-morphisms on $\mathrm{Diff}_0(S, \mathrm{area})$ whose absolute values bound from below the topological entropy. If time permits, I will define a bi-invariant metric on this group, called the entropy metric, and show that it is unbounded. Based on a joint work with M. Marcinkowski.
A group $G$ satisfies the Tits alternative if each of its finitely generated subgroups contains a non-abelian free group or is virtually solvable. I will sketch a proof of a theorem saying that if $G$ acts geometrically on a simply connected nonpositively curved complex built of equilateral triangles, then it satisfies the Tits alternative. This is joint work with Damian Osajda.
Random walks are not new to geometric group theory (see, for example, work of Furstenberg, Kaimanovich, Masur). However, following independent proofs by Maher and Rivin that pseudo-Anosovs are generic within mapping class groups, and then new techniques developed by Maher-Tiozzo, Sisto, and others, the field has seen in the past decade a veritable explosion of results. In a 2-paper series, we answer with fine detail a question posed by Handel-Mosher asking about invariants of generic outer automorphisms of free groups and then a question posed by Bestvina as to properties of $\mathbb{R}$-trees of full hitting measure in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich, Joseph Maher, and Samuel J. Taylor.
Given a group $G$ acting properly by isometries on a metric space $X$, the exponential growth rate of $G$ with respect to $X$ measures "how big" the orbits of $G$ are. If $H$ is a subgroup of $G$, its exponential growth rate is bounded above by the one of $G$. We are interested in the following question: when do $H$ and $G$ have the same exponential growth rate?
This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80s that a group $Q = F/N$ (written as a quotient of the free group) is amenable if and only if $N$ and $F$ have the same exponential growth rate (with respect to the word length in $F$). About the same time, Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogous to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics, and group theory.
In this talk, we are interested in the case where $G$ acts on an arbitrary Gromov hyperbolic space and propose a framework that encompasses both the combinatorial and the geometric point of view. We will see that as soon as the action of $G$ on $X$ is "reasonable" (proper co-compact, cuspidal with parabolic gap, or more generally strongly positively recurrent), then $G$ and $H$ have the same growth rate if and only if $H$ is co-amenable in $G$. Our strategy is based on a new kind of Patterson-Sullivan measure taking values in a space of bounded operators.
This is joint work with R. Dougall, B. Schapira, and S. Tapie.
For a word-hyperbolic group $G$, the notion of quasiconvexity is independent of the choice of a generating set and a quasiconvex subgroup of $G$ is quasi-isometrically embedded in $G$. Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. However, beyond word-hyperbolic groups, the notion of quasiconvexity is not as useful. For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group: a "stable" subgroup and a "Morse" subgroup. In this talk, we will discuss various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups.
Despite the simple looking presentation much is unknown about Artin groups. However, some questions can be answered in suitable classes of Artin groups. I will discuss the residual finiteness of Artin groups of large type on three generators. This relies on splittings of such Artin groups as amalgamated products of finitely generated free groups.
A Helly graph is a graph in which the metric balls form a Helly family: any pairwise intersecting collection of balls has nonempty total intersection. A Helly group is a group that acts properly and cocompactly on a Helly graph. Helly groups simultaneously generalize hyperbolic, cocompactly cubulated, and C(4)-T(4) graphical small cancellation groups while maintaining nice properties, such as biautomaticity. I will show that if a crystallographic group is Helly then its point group preserves an $L^{\infty}$ metric on $\mathbb{R}^n$. Thus we will obtain some new nonexamples of Helly groups, including the 3-3-3 Coxeter group, which is a systolic group. This answers a question posed by Chepoi during the recent Simons Semester on Geometric and Analytic Group Theory in Warsaw.
Whitehead's asphericity conjecture states that a subcomplex of an aspherical 2-complex is aspherical. He formulated it as a question in 1941. Whitehead noticed that an affirmative answer implies the asphericity of knot complements, a fact unknown at the time. Papakyriakopoulos established the asphericity of knot complements in 1957 using 3-manifold topology and not relying on Whitehead's ideas. The Whitehead conjecture remains unresolved to this day. Generalized knot complements and their 2-dimensional spines remain at the center of the conjecture. Presentations for such spines can be encoded by labeled oriented trees, lot's for short. In my talk I will give a survey of the algebra, topology, and geometry surrounding labeled oriented trees.
When $1 \rightarrow H \rightarrow G \rightarrow Q \rightarrow 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an "ending lamination" on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of $\mathrm{Out}(F_n)$, one can identify the resultant quotient space with a certain $\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.
I will give a quick introduction to $L^2$-Betti numbers and describe a result explaining why the second $L^2$-Betti number vanishes for certain compact 2-complexes whose fundamental groups are special, in the sense that they are subgroups of right-angled Artin groups.
Gersten and Short showed that given a quasiconvex subgroup $H$ of a hyperbolic group $G$, for any finite generating set $S$ of $G$, the language of geodesics in $\mathrm{Cay}(G,S)$ representing elements of $H$ is a regular language. Having a regular language of geodesics is typically useful for understanding the growth function of the respective group/subgroup. For example, Dahmani, Futer, and Wise make use of the above fact to show that non-elementary hyperbolic groups grow exponentially more quickly than their infinite-index quasi-convex subgroups.
A group is said to be MLTG if local Morse quasi-geodesics are global. The class of MLTG include mapping class groups, CAT(0) groups, Teichmuller spaces, graph products of hyperbolic groups, and a large class of hierarchically hyperbolic groups.
Stable subgroups of finitely generated groups generalize quasi-convex subgroups of hyperbolic groups, and if the group $G$ is hyperbolic, then the two notions coincide.
We show that given a stable subgroup $H$ of some MLTG group $G$, for any finite generating set $S$ of $G$, the language of geodesics representing elements of $H$ is a regular language. As an application, and in the spirit of Dahmani, Futer, and Wise's result above, we show that torsion-free MLTG groups grow exponentially more quickly than their infinite-index stable subgroups. The talk is based on an ongoing project with Cordes, Russell, and Spriano.
A graph is Helly if every family of pairwise intersecting (combinatorial) balls has common intersection. Groups acting geometrically - that is, properly and cocompactly - on Helly graphs are themselves called Helly. Such graphs and groups possess various non-positive-curvature-like features. Moreover, Helly graphs are closely related to injective metric spaces, whose behavior is very similar to CAT(0) spaces, and Helly groups act geometrically on injective metric spaces as well. In the talk I will overview the main examples of Helly groups and their important properties. The talk is based on works with Jeremie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, and Jingyin Huang.
Given $\varphi \colon F_n \to F_n$ an automorphism of a free group of rank $n$, there is an associated free-by-cyclic group $F_n \rtimes_\varphi \mathbb{Z}$, which may be thought of as the mapping torus of the automorphism. Properties of the automorphism determine properties of the mapping torus and vice-versa. Gersten gave a simple example $\psi \colon F_3 \to F_3$ of an automorphism whose mapping torus is a "poison subgroup" for nonpositive curvature, in the sense that any group containing $F_3 \rtimes_\psi \mathbb{Z}$ is not a CAT(0) group. In the opposite direction, Hagen-Wise and Button-Kropholler proved certain families of automorphisms have mapping tori that are cocompactly cubulated. We prove that a large class of polynomially-growing free group automorphisms admitting an additional symmetry have CAT(0) mapping tori. The key tool is a representation of these automorphisms as relative train track maps on orbigraphs, certain graphs of groups thought of as orbi-spaces. This gives a hierarchy for the mapping torus. It is an interesting question whether or not our mapping tori are cocompactly cubulated.
Circularly-orderable and left-orderable groups play an important, and sometimes surprising role in low-dimensional topology. Motivated by these connections, I will present new necessary and sufficient conditions for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. This raises a plethora of intriguing questions, especially when the group is the fundamental group of a manifold. Time permitting, I will talk about how this work leads to new progress in answering the question of when the direct product of two circularly-orderable groups is circularly-orderable.
This is joint work with Jason Bell and Adam Clay.
Let $M$ be a closed Riemannian manifold and let $\mu$ be the measure induced by the volume form. Denote by $\mathrm{Homeo}_0(M, \mu)$ the group of all $\mu$-preserving homeomorphisms of $M$ isotopic to the identity. It is well-known that the second bounded cohomology of $\mathrm{Diff}_0(M, \mu)$ is infinite-dimensional due to existence of quasimorphisms on $\mathrm{Diff}_0(M, \mu)$ (Gambaudo-Ghys, Polterovich). In this talk, I will explain how to construct bounded classes in higher dimensions. As an application, we will show that under certain conditions on the fundamental group of $M$, the third bounded cohomology of $\mathrm{Diff}_0(M, \mu)$ is infinite-dimensional. If time permits, I will discuss how this construction can be used to construct invariants of foliated fibre bundles. It is a joint work with Michael Brandenbursky and Martin Nitsche.
A natural notion of complexity for a closed manifold $M$ is the smallest number of top dimensional simplices it takes to triangulate $M$. Gromov showed that a variant of this notion called simplicial volume gives a lower bound for the volume of $M$ with respect to any (normalized) Riemannian metric. The heart of his proof factors through the dual notion of bounded cohomology. I will define bounded cohomology of discrete groups illustrated by some examples coming from computing the volumes of geodesic simplices in hyperbolic space. Although bounded cohomology is often an unwieldy object evading computation, we give some conditions for volume classes to be non-vanishing in low dimensions. We then ask, "When do higher dimensional volume classes vanish?"
It is known that, under mild assumptions, for a free group $F_r$ of finite rank $r>2$, a "random" element $\phi_n\in \mathrm{Out}(F_r)$, obtained after $n$ steps of a random walk on $\mathrm{Out}(F_r)$, is fully irreducible (a free group analog of being pseudoAnosov), and that an a.e. trajectory of the walk converges to a point in the boundary of the Culler-Vogtmann Outer space $CV_r$. We prove that generically the attracting $\mathbb R$-tree $T_+(\phi_n)\in \partial CV_r$ for such a random fully irreducible $\phi_n$ is trivalent (that is, all branch points of $T_+$ have degree 3) and nongeometric, (that is $T_+$ is not the dual tree of any measured foliation of a finite 2-complex).
Similarly, for the exit/harmonic measure $\nu$ of the random walk on the boundary $\partial CV_r$ of the Outer space, we prove that a $\nu$-a.e. $\mathbb R$-tree $T\in \partial CV_r$ is trivalent and nongeometric. The talk is based on joint work with Joseph Maher, Catherine Pfaff and Samuel Taylor.
Given a hyperbolic space, one can define a topological space known as the boundary at infinity. Since the boundary is invariant under quasi-isoemtries, it provides us with an invariant of hyperbolic groups up to quasi-isometry. Generalizing this idea to non-hyperbolic groups has proven difficult for various reasons. In recent years, the Morse boundary has emerged as a promising generalisation, although there are several open questions regarding its topology and metrizability.
In this talk, we will review the Morse boundary and several of the topologies it carries. We will compare these topologies and illustrate how they are markedly different from each other, already when considering Morse boundaries of CAT(0) cube complexes. We will show how the extra structure of CAT(0) cube complexes helps to construct a metric on the Morse boundary and how it can shed some light on quasi-isometry invariance.
The simplicial volume of a simplicial complex is a topological invariant related to the growth of the fundamental group, which gives rise to a semi-norm in homology. In this talk, we introduce the volume entropy semi-norm, which is directly related to the growth of the fundamental group of simplicial complexes and shares functorial properties with the simplicial volume. Answering a question of Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. We also raise a few questions where the growth of a finitely presented group and its asymptotic growth interact. Joint work with I. Babenko.
I will discuss some known results about faithful representations of finitely generated groups (in particular, surface and 3-manifold groups) into compact Lie groups.