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Seminar organizers: Sami Douba, Piotr Przytycki

Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.

It is known since the late 70s that in locally symmetric spaces of large injectivity radius, the $k$-th real Betti number divided by the volume is approximately equal to the $k$-th $L^2$-Betti number. Is there an analogue of this fact for mod-$p$ Betti numbers? This question is still very far from being solved, except for certain special families of locally symmetric spaces. In this talk, I want to advertise a relatively new approach to study the growth of mod-$p$ Betti numbers based on a quantitative description of minimal area representatives of mod-$p$ homology classes.

I'll describe an effective algorithm to build grids in many metric spaces of interest in geometric group theory, e.g. locally symmetric spaces. Joint work with Aurel Page.

A slope $p/q$ is a characterizing slope for a knot $K$ in the $3$-sphere if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. It is known that for a given torus knot all but finitely many non-integer slopes are characterizing and that for hyperbolic knots all but finitely many slopes with $q>2$ are characterizing. I will discuss the proofs of these results, which have a surprising amount in common.

Being of type $FP_2$ is an algebraic shadow of being finitely presented. A long standing question was whether these two classes are equivalent. This was shown to be false in the work of Bestvina and Brady. More recently, there are many new examples of groups of type $FP_2$ coming with various interesting properties. I will begin with an introduction to the finiteness property $FP_2$. I will end by giving a construction to find groups that are of type $FP_2(\mathbb{F})$ for all fields $\mathbb{F}$ but not $FP_2(\mathbb{Z})$.

For any hierarchically hyperbolic group, and in particular any mapping class group, we define a new metric that satisfies a coarse Helly property. This enables us to deduce that the group is semihyperbolic, i.e. that it admits a bounded quasigeodesic bicombing, and also that it has finitely many conjugacy classes of finite subgroups. This has several other consequences for the group. This is a joint work with Nima Hoda and Harry Petyt.

Mapping class groups of surfaces of genus at least 3 are perfect, but their finite-index subgroups need not be—they can have non-trivial abelianizations. A well-known conjecture of Ivanov states that a finite-index subgroup of a mapping class group of a sufficiently high genus has finite abelianization. We will discuss a proof of this conjecture, which goes through an equivalent representation-theoretic form of the conjecture due to Putman and Wieland.

We give a new proof of a theorem of Koberda which says that right-angled Artin subgroups of mapping class groups abound. This alternative approach uses the hierarchical structure of the curve complex, which allows for more explicit computations. Time permitting, we will also discuss some applications to the theory of convex cocompactness in mapping class groups.

A hyperbolic group acts naturally by homeomorphisms on its boundary. The theme of this talk is to say that, in many cases, such an action has very robust dynamics.

Jonathan Bowden and I studied a very special case of this, showing if G is the fundamental group of a compact, negatively curved Riemannian manifold, then the action of G on its boundary is topologically stable (small perturbations of it are semi-conjugate, containing all the dynamical information of the original action). In new work with Jason Manning, we get rid of the Riemannian geometry and show that such a result holds for hyperbolic groups with sphere boundary, using purely large-scale geometric techniques.

This theme of studying topological dynamics of boundary actions dates back at least as far as work of Sullivan in the 1980's, although we take a very different approach. My talk will give some history and some picture of the large-scale geometry involved in our work.

It is still an open problem whether or not Coxeter groups are biautomatic. A 2-dimensional Coxeter group is a Coxeter group whose finite parabolic subgroups are all dihedral groups. Damian Osajda, Piotr Przytycki, and I were able to prove that 2-dimensional Coxeter groups are biautomatic. In this talk, I will present the outline of our proof and some of the difficulties of generalizing it to other classes of Coxeter groups. The talk will be largely self-contained, although previous exposure to Coxeter groups and biautomaticity will of course be helpful.

Let $S$ be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of the orbit equivalence relation coming from the action of the mapping class group of $S$ on the Gromov boundary of the arc graph of $S$. This is joint work with Marcin Sabok.

Convex co-compact representations are a generalization of convex co-compact Kleinian groups. A discrete faithful representation into the projective linear group is called convex co-compact if its image acts co-compactly on a properly convex domain in real projective space. In this talk, I will discuss such representations of 3-manifold groups. I will prove that a closed irreducible orientable 3-manifold group admits such a representation only when the manifold is geometric (with Euclidean, hyperbolic, or Euclidean $\times$ hyperbolic geometry) or when each component in its geometric decomposition is hyperbolic. This extends a result of Benoist about convex real projective structures on closed 3-manifolds. In each case, I will also describe the structure of the representation and the properly convex domain. This is joint work with Andrew Zimmer.

A random group in the triangular binomial model $\Gamma(n,p)$ is given by the presentation $\langle S|R \rangle$, where $S$ is a set of $n$ generators and $R$ is a random set of cyclically reduced relators of length 3 over $S$, with each relator included in $R$ independently with probability $p$. When $n\rightarrow\infty$, the asymptotic properties of groups in $\Gamma(n,p)$ vary widely with the choice of $p=p(n)$. By Antoniuk-Łuczak-Świątkowski and Żuk, there exist constants $C, C'$, such that a random triangular group is asymptotically almost surely (a.a.s.) free, if $p < Cn^{-2}$, and a.a.s. infinite, hyperbolic, but not free, if $p\in (C'n^{-2}, n^{-3/2-\varepsilon})$. We generalize the second statement by finding a constant $c$ such that, if $p\in(cn^{-2}, n^{-3/2-\varepsilon})$, then a random triangular group is a.a.s. not left-orderable. We prove this by linking left-orderability of $\Gamma \in \Gamma(n,p)$ to the satisfiability of a random propositional formula, constructed from the presentation of $\Gamma$. The left-orderability of quotients will be also discussed.

Consider a triangle in the Euclidean plane and subdivide it recursively into 4 sub-triangles by joining its midpoints. Each generation of this iterated subdivision yields triangles which are all similar to the original one and exactly twice as small as the triangle(s) of the previous generation. What happens when we perform this iterated medial triangle subdivision on a geodesic triangle when the underlying space is not Euclidean? I will first produce examples of various unfamiliar and unexpected behaviours of this subdivision in non-Euclidean geometries. I will then show that this iterated subdivision nevertheless "stabilizes in the limit" (in a sense that will be made precise) when the underlying space is of constant non-zero curvature. My aim is to combine this result with a forthcoming result of Christopher Bishop on conforming triangulations of PSLGs to construct acute triangulations of hyperbolic and spherical simplicial complexes.

A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free measurable actions of these groups on a standard probability space. In this vein, the amenable groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation on a standard probability space, deriving the analogues of structural properties of amenable subgroups (copies of $\mathbb{Z}$) of a free group. Most importantly, just like every such subgroup is contained in a unique maximal one, we show that even in the non-measure-preserving setting, every hyperfinite subequivalence relation is contained in a unique maximal one.

In studying moduli spaces of representations of surface groups, and more generally of hyperbolic groups, triangle groups are simple examples which can provide insight into the more general theory. Recent work of Alessandrini–Lee–Schaffhauser generalized the theory of higher Teichmüller spaces to the setting of orbifold surfaces, including triangle groups. In particular, they defined a "Hitchin component" of representations into $\mathrm{PGL}(n,\mathbb{R})$ which is homeomorphic to a ball and consists entirely of discrete and faithful representations. They compute the dimension of Hitchin components for triangle groups, and find that this dimension is positive except for a finite number of low-dimensional examples where the representations are rigid. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with a diagonal embedding into $\mathrm{PGL}(2n,\mathbb{R})$ or $\mathrm{PSp}(2n,\mathbb{R})$ is always locally rigid.

The Cremona group is the group of birational transformations of the projective plane. Even if this group comes from algebraic geometry, tools from geometric group theory have been powerful to study it. In this talk, based on a joint work with Christian Urech, we will build a natural action of the Cremona group on a CAT(0) cube complex. We will then explain how we can obtain new and old group theoretical and dynamical results on the Cremona group.

The notion of measure equivalence between countable groups was introduced by Gromov as a measure-theoretic analogue of quasi-isometry. We study the class of 2-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and show that if two groups from this class are measure equivalent, then their "curve graphs" are isomorphic. This reduces the question of measure equivalence of these groups to a combinatorial rigidity question concerning their curve graphs; in particular, we deduce measure equivalence superrigidity results for a class of Artin groups whose curve graphs are known to be rigid from a previous work of Crisp. There are two main ingredients in the proof of independent interest. The first is a more general result concerning boundary amenability of groups acting on certain CAT($-1$) spaces. The second is a structural similarity between these Artin groups and mapping class groups from the viewpoint of measure equivalence. This is joint work with Camille Horbez.

In 2001, Crisp and Paris showed the squares of the standard generators of an Artin group generate an "obvious" right-angled Artin subgroup. This resolved an earlier conjecture of Tits. I will introduce a generalization of this conjecture, where we ask that a larger set of elements generates another "obvious" right-angled Artin subgroup. I will give evidence that this is a good generalization, explain what classes of Artin groups we can prove it for, and give some applications. Joint with Kasia Jankiewicz.

A *complete square complex* is a 2-complex $X$ whose universal cover is the product of two trees. Obvious examples are when $X$ is itself the product of two graphs but there are many other examples. I will give a quick survey of complete square complexes with an aim towards describing some problems about them and describing some small examples that are "irreducible" in the sense that they do not have a finite cover that is a product.

The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years. In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, $\mathrm{Out}(F_n)$, and right-angled Artin and Coxeter groups, among many others. I will discuss how a random element of such a group interacts with fixed subgroups, especially so-called hyperbolically embedded subgroups. In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.

Suppose a group $G$ contains an infinite-order element $g$ such that every finite-dimensional linear representation of $G$ maps some nontrivial power of $g$ to a unipotent matrix. Since unitary matrices are diagonalizable, and since a unipotent matrix is torsion if its entries lie in a field of positive characteristic, such a group $G$ does not admit a faithful finite-dimensional unitary representation, nor is $G$ linear over a field of positive characteristic. We discuss manifestations of the above phenomenon in various finitely generated groups, with an emphasis on 3-manifold groups.

Ian Agol showed that hyperbolic groups acting geometrically on CAT(0) cube complexes are virtually special in the sense of Haglund–Wise, the last step in the proof of the virtual Haken and virtual fibering conjectures. I will talk about a generalization of this result (also obtained independently by Groves and Manning), which states that cubulated relatively hyperbolic groups are virtually special provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. In particular, we deduce virtual specialness for cubulated groups that are hyperbolic relative to virtually abelian groups, extending Wise's results for limit groups and fundamental groups of cusped hyperbolic 3-manifolds. The main ingredient of the proof is a relative version of Wise's quasi-convex hierarchy theorem, obtained using recent results by Einstein, Groves, and Manning.

The geometrization theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of only eight homogeneous 3-dimensional Riemannian model spaces, called the Thurston geometries. So to begin to understand what 3-manifolds "are like," we may reduce the problem to first understanding these geometric pieces.

For me, the happy fact that our day-to-day life takes place in three dimensions is a major asset here; while we can visualize surfaces extrinsically, and reason about 4-manifolds via slicing, only for 3-manifolds can we really attempt to answer "what would it feel like/look like/be like" to live inside of one. To leverage our natural visual intuition in three dimensions, in joint work with Rémi Coulon, Sabetta Matsumoto, and Henry Segerman, we have adapted the computer graphics technique of raymarching to homogeneous Riemannian metrics. We use this to produce accurate and real-time intrinsic views of Riemannian 3-manifolds; specifically the eight Thurston geometries and assorted compact quotients. In this talk, I will take you on a tour of these spaces, and talk a bit about the mathematical challenges of actually implementing this.

A topological dynamical system (i.e. a group acting by homeomorphisms on a compact Hausdorff space) is said to be proximal if for any two points $p$ and $q$ we can simultaneously "push them together" (rigorously, there is a net $g_n$ such that $\lim g_n(p) = \lim g_n(q)$). In his paper introducing the concept of proximality, Glasner noted that whenever $\mathbb{Z}$ acts proximally, that action will have a fixed point. He termed groups with this fixed point property “strongly amenable”. The Poisson boundary of a random walk on a group is a measure space that corresponds to the space of different asymptotic trajectories that the random walk might take. Given a group $G$ and a probability measure $\mu$ on $G$, the Poisson boundary is trivial (i.e. has no non-trivial events) if and only if $G$ supports a bounded $\mu$-harmonic function. A group is called Choquet–Deny if its Poisson boundary is trivial for every $\mu$.

In this talk I will discuss work giving an explicit classification of which groups are Choquet–Deny, which groups are strongly amenable, and what these mysteriously equivalent classes of groups have to do with the ICC property. I will also discuss why strongly amenable groups can be viewed as strengthening amenability in at least three distinct ways, thus proving the name is extremely well deserved. This is joint work with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowsi.