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

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.

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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.