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

The talk is based on two recent preprints:

- [GGPY], I. Gekhtman, V. Gerasimov, L.P., W. Yang,
*Martin boundary covers Floyd boundary* - [DGGP], M. Dussaule, I. Gekhtman, V. Gerasimov, L.P.,
*The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups*

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.

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.

Celebrated work of Eels-Sampson and Hartman asserts the existence of a harmonic diffeomorphism in any homotopy class of maps between a pair of homeomorphic compact hyperbolic surfaces. The study of harmonic maps has since been vastly generalized in the work of Gromov-Schoen, Korevaar-Schoen, and Jost, with wide-ranging application. Motivated by the nonabelian Hodge correspondence, we pursue a discrete version of the theory that is more accessible computationally; in particular our main theorem is that, in the setting of compact hyperbolic surfaces, the discrete energy functional is strongly convex (a property not known for the smooth energy functional). These ideas are implemented in a user-friendly computer program that I will present. This is joint work with Brice Loustau and Léonard Monsaingeon.

I will motivate and describe the construction of an infinite family of cells called bisimplices. The 1-skeleta of these cells are complete bipartite graphs, making them well suited to the construction of higher skeleta of bipartite graphs. The construction of bisimplices poses special challenges since, unlike simplices and cubes, they cannot be realized as polyhedra. An essential component of the construction is an application of the discrete Morse theory of Forman to the recognition of spheres.

I'll describe a relationship between the smallest 1-form Laplacian eigenvalue $\lambda_1^*$ and surface complexity on hyperbolic manifolds. Using this relationship, we'll discuss "surface theft", a prospect for proving good lower bounds on $\lambda_1^*(M)$ for big congruence arithmetic hyperbolic 3-manifolds $M$. This is joint work with Mark Stern.

The "spelling theorem" is a classical result about disk diagrams for one-relator groups with torsion $\langle a,b \> | \> W^n\rangle$. But how to generalize this striking result? We introduce the notion of a bicollapsible 2-complex. This allows us to generalize the hyperbolicity of one-relator groups with torsion to a broader class of groups with presentations whose relators are proper powers. We also prove that many such groups act properly and cocompactly on a CAT(0) cube complex. This is joint work with Jonah Gaster.

Coxeter groups with large enough torsion coefficients are known to satisfy non-positive curvature properties such as systolicity. However, there are known examples of Coxeter groups which are not systolic. In particular, the associated Coxeter complexes are not systolic. For a class of such Coxeter groups, we prove a generalized non-positive curvature condition—weak modularity. Time permitting, we extend the result to buildings.

Parking functions are certain simple combinatorial objects which play an important role in the study of symmetric functions. I will give a very brief indication of their importance, but my main topic will be a new characterization of parking functions. A (rational) parking function can be encoded as a word of length $n$ on the alphabet $\{0, \dots, m-1\}$, but not all words correspond to parking functions. To any word, we associate a piecewise linear transformation of $\mathbb{R}^m$. We show that this transformation has a fixed point if and only if the word corresponds to a parking function. This is useful because it allows us to prove that a certain map (usually called "zeta") from parking functions to parking functions, defined when $m$ and $n$ are relatively prime, is in fact a bijection, verifying a conjecture of Gorsky, Mazin, and Vazirani. Perhaps more relevantly to the audience, the geometry of these piecewise-linear transformations seems as if it may also contain further interesting information. This talk is based on joint work with Jon McCammond and Nathan Williams.

A 2-system of arcs on an $n$-punctured sphere $S$ is a collection of (homotopy classes of) simple arcs on $S$ joining punctures and pairwise intersecting at most twice. Bar-Natan proved that a 2-system of arcs on $S$ beginning and ending at a fixed puncture has size at most $\binom{n}{3}$. In this talk, I will sketch a proof that the same holds for a 2-system of arcs on $S$ joining a fixed pair of distinct punctures.

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the classical groups U($n$), O($n$), or Sp($n$). The group structure of these matrices allows us to go further and find surprising algebraic quantities hidden in the values of these integrals. The talk will be aimed at a wide audience, and all notions will be explained. Based on joint work with Michael Magee (Durham).

We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then the action is trivial. In particular, all actions of finitely generated torsion groups on such complexes are trivial. As an ingredient, we prove that the image of an immersed loop in a graph of girth $2\pi$ with length not commensurable to $\pi$ has diameter $>\pi$. This is related to a theorem of Dehn on tiling rectangles by squares. Joint work with Sergey Norin and Damian Osajda.

A group is said to satisfy the Tits Alternative if its finitely generated subgroups exhibit a striking dichotomy: they are either "very large" (they contain a non-abelian free subgroup) or "very small" (they are virtually soluble). Many groups of geometric interest have been shown to satisfy the Tits Alternative: linear groups, mapping class groups of hyperbolic surfaces, etc. In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

We discuss Thurston's universal circle representation and describe how it can be used to study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. This is joint work with Ying Hu.

How many simple closed curves can you draw on the surface of genus $g$ in such a way that no two are isotopic and no two intersect in more than $k$ points? It is known how to draw a collection in which the number of curves grows as a polynomial in $g$ of degree $k+1$, and conjecturally, this is the best possible. I will describe a proof of an upper bound that matches this function up to a factor of $\log(g)$. It is based on an elegant geometric argument due to Przytycki and employs some novel ideas blending covering spaces and probabilistic combinatorics.