Seminar organizers: Macarena Covadonga Robles Arenas, Sami Douba, Nima Hoda, Piotr Przytycki, Daniel Wise

I will introduce the operation, called *dense amalgam*,
which to any tuple $X_1,\dots,X_k$ of non-empty compact metric
spaces associates some disconnected perfect compact metric space,
denoted $\widetilde\sqcup(X_1,\dots,X_k)$, in which there are many
appropriately distributed copies of the spaces $X_1,\dots,X_k$. I
will also present a convenient characterization of dense amalgams,
in terms of a list of properties, similar in spirit to the well
known characterization of the Cantor set. I will explain that, in
various settings, the ideal boundary of the free product of groups
(amalgamated along finite subgroups) is homeomorphic to the dense
amalgam of boundaries of the factors. For example, the boundary
of a Coxeter group which has infinitely many ends, and which is
not virtually free, is the dense amalgam of the boundaries of the
maximal 1-ended special subgroups.

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This is joint work with Jingyin Huang (OSU). We define a metrically systolic complex as flag simplicial complex with a given metric on their two-skeleton satisfying some local conditions. I will present few features of such complexes resembling the ones of non-positively curved spaces. Then I will talk about two-dimensional Artin groups, metrically systolic complexes they act on, and consequences of admitting such actions. One of them is solvability of the conjugacy problem for Artin groups in question and for all their finitely presented subgroups.

The Borel chromatic number – introduced by Kechris, Solecki, and Todorcevic (1999) – generalizes the chromatic number on finite graphs to definable graphs on topological spaces. We will see examples of graphs with chromatic number $2$ which cannot be colored in a Borel way with less than $3$ colors (or even any finite number of colors). Many interesting examples of Borel graphs also arise from the continuous or Borel action of a finitely generated group on a topological space. I will explain the proof that a Borel graph with bounded degree $k$ can always be colored using $k+1$ colors in a Borel way. Finally I will discuss the problem of characterizing the Borel graphs with finite Borel chromatic number.

In the plane any obtuse triangle can be dissected into 2
right-angled triangles by drawing the altitude from the obtuse
vertex on its opposite side. The problem generalises naturally to
higher dimensions, where we consider a simplex to be
*non-obtuse* provided no angle between two co-dimension 1
faces is obtuse. The problem at hand is then stated in an open
conjecture due to Hadwiger (1956): can every $d$-simplex be
dissected into non-obtuse $d$-simplices? We provide a constructive
answer to this interesting puzzle in dimension 3, where we show
that 28 orthoschemes are sufficient to dissect any
tetrahedron.

We study groups of isometries of $\delta$-hyperbolic spaces which are not necessarily proper. Such examples occur often in geometry and topology, most notably the action of the mapping class group on the curve complex, or of $\operatorname{Out}(F_n)$ on the free factor complex. Since the action is not proper, one needs a weak properness condition, namely the WPD condition formulated by Bestvina-Fujiwara. Under this condition, we prove that for random walks on isometry groups of $\delta$-hyperbolic spaces generic elements are WPD, and the normal closure of a generic element is a free group. This answers a question of D. Margalit for the mapping class group, and for the Cremona group gives a new proof of the abundance of normal subgroups due to Cantat-Lamy. Joint with Joseph Maher.

Leighton's graph covering theorem states that two finite graphs with isomorphic universal covers have isomorphic finite covers. I will discuss a new proof that involves using the Haar measure to solve a set of gluing equations. I will discuss generalizations to graphs with fins, and applications to quasi-isometric rigidity.

Artin groups arise naturally as fundamental groups of hyperplane complements. They form a large and diverse collection of groups which include braid groups, free groups, free abelian groups and many more. While some classes of Artin groups are well understood, many remain mysterious, with even very basic questions unanswered. In this talk I will review what is known and not known, then discuss some geometric techniques for studying these groups.

It follows from a well-known theorem of Peter Scott that each immersed curve with minimal self-intersections on a surface $S$ lifts as an embedded curve to some finite covering of $S$. In recent years, there have been some efforts to bound the minimum degree of these coverings. I will talk about some results obtained in this direction.

Let $G$ be a finitely generated group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G.$ A natural question is the following: what are the possible asymptotic behaviours for the length of $f^n(g)$, written as a word in the generating set $S$, as $n$ goes to infinity, and as $g$ varies in the group $G$? Growth was described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We investigate the case of a general torsion-free hyperbolic group. This is a joint work with Rémi Coulon, Arnaud Hilion, and Gilbert Levitt.

During the talk, I will discuss a problem concerning extensions of partial isomorphisms of finite tournaments and its equivalent form, which is a problem about the pro-odd topology on the free group. The pro-odd topology is the refinement of the profinite topology where we take only the normal subgroups of odd index as the neighborhood basis of 1. The problem concerns a characterization of those finitely generated subgroups of the free group which are closed in the pro-odd topology. I will discuss a positive answer for cyclic subgroups and a negative answer for a generalized version of this question. Joint work with J. Huang, M. Pawliuk, and D. Wise.

Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups in a given group $G$. They arise as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting.

Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 symmetric space, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. If the subgroup is of divergence type, we show its critical exponent is in fact equal to the dimension of the boundary. If $G$ has property (T) we obtain as a corollary that an IRS of divergence type must in fact be a lattice. The proof uses ergodic theorems for actions of hyperbolic groups.

I will also talk about results about growth rates of normal subgroups of hyperbolic groups that inspired this work.

Buildings were originally introduced by Jacques Tits around the 1950s as a geometric tool to study the semisimple Lie (and algebraic) groups. By now, buildings have a very rich theory of their own; for instance, they possess a "non-positively curved" cellular complex structure, which makes them prominent objects of interest in geometric group theory. In this talk, I will investigate certain combinatorial analogues of geodesic rays in buildings (the "geodesic ray bundles") and give some intuition about how they look like. Using recent results of Huang, Sabok and Shinko, I will then derive some consequences on the topic of hyperfinite group actions.

The polygonalization complex of a punctured surface is a cube complex whose vertices correspond to polygonal decompositions of the surface. Although the complex is not CAT(0), it is contractible and shares many features of a CAT(0) cube complex. In particular, it has a very rich family of separating and embedded hyperplanes. We will parametrize it and characterize its crossing graph via the arc complex. This is a joint work with M. Bell and R. Tang.

For an infinite discrete group $G$, the classifying space for proper actions $\underline{E}G$ is a proper $G$-CW-complex $X$ such that for every finite subgroup $F \subset G$ the fixed point set $X^F$ is contractible. In joint work with Nansen Petrosyan we describe a procedure of constructing new models for $\underline{E}G$ out of the standard ones, provided the action of $G$ on $\underline{E}G$ admits a strict fundamental domain. Our construction is of combinatorial nature, and it depends only on the structure of the fundamental domain. The resulting model is often much "smaller" than the old one, and thus it is well-suited for (co-)homological computations. Before outlining the construction, I shall give some background on the space $\underline{E}G$. I will also discuss some examples and applications in the context of Coxeter groups, graph products of finite groups, and automorphism groups of buildings.

In the hyperbolic setting, groups with Menger curve boundary are known to be abundant. It was a surprising observation of Ruane that there were no known examples of non-hyperbolic groups with Menger curve boundary found in the literature. Thus Ruane posed the problem (early 2000's) of finding examples (alt. interesting classes) of non-hyperbolic groups with Menger curve boundary. In this talk I will discuss the first class of such examples. This is joint work with Chris Hruska and Bakul Sathaye. Time permitting, I will also discuss related work concerning right-angled Coxeter groups.

A finitely generated group is strongly rigid if any self quasi-isometry of the group is uniformly close to an automorphism of the group. We show many 2-dimensional Artin groups are strongly rigid. This is joint work with D. Osajda.