Seminar organizers: William Chong, Christopher Karpinski, Zachary Munro, Piotr Przytycki, Daniel Wise

A (strict) bramble in a graph G is a collection of subgraphs of G such that the union of any number of them is connected. The order of a bramble is the smallest size of a set of vertices that intersects each of the subgraphs in it. Brambles have long been part of the graph minor theory toolkit, in particular, because a bramble of high order is an obstruction to existence of a low width tree decomposition. We will discuss high dimensional analogues of brambles. In particular, we show that an d-dimensional bramble of high order in a d-dimensional simplicial complex X is an obstruction to existence of a low multiplicity continuous map from X to R^d (and more generally to any d-dimensional contractible complex). This can be seen as a qualitative variant of Gromov's topological overlap theorem. As an application, we construct the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. Based in part on joint work with David Eppstein, Robert Hickingbotham, Laura Merker, Michał T. Seweryn and David R. Wood.

Artin groups are a family of groups generalizing braid groups and closely related to Coxeter groups. They can be realized as the fundamental groups of certain complex hyperplane arrangements, which conjecturally are their $K(\pi,1)$ spaces. This is known as the $K(\pi,1)$ conjecture. There is also a conjectural description of the center of every Artin group. Irreducible Artin groups, i.e. those that do not split as direct products, are conjectured to have trivial centers, unless they are of finite type, in which case they are known to have infinite cyclic centers. In my talk, I will present joint work with Kevin Schreve, where we show that the Artin groups satisfying the $K(\pi,1)$ conjecture also satisfy the center conjecture.

I will introduce the cactus group, a relative of the braid group. Like its more famous cousin, it appears naturally in representation theory and is the fundamental group of a very natural space. I will describe this space, called the moduli space of genus 0 real curves. Then I will discuss a variant of the cactus group, which we call the virtual cactus group. Along the way, we will see the solution to the following question: what is the space of solutions to the equation $a + b = c$, where $a, b, c$ are elements of $\mathbb{R}P^1$?

We show that groups acting properly and cocompactly by type-preserving automorphisms on buildings satisfy a weak Tits alternative: they are either virtually abelian or contain a non-abelian free subgroup. This is joint work with Damian Osajda and Piotr Przytycki.

A pseudo-Anosov map is a surface homeomorphism that acts with similar dynamics as a hyperbolic element of $\mathrm{SL}_2 \mathbb{R}$ on $\mathbb{R}^2$. A classical result of Nielsen and Thurston shows that these are surprisingly prevalent among mapping classes of surfaces. The dilatation of a pseudo-Anosov map is a measure of the complexity of its dynamics. It is another classical result that the set of dilatations among all pseudo-Anosov maps defined on a fixed surface has a minimum element. This minimum dilatation can be thought of as the smallest amount of mixing one can perform on the surface while still doing something topologically interesting. The minimum dilatation problem asks for this minimum value. In this talk, we will start by providing some background for pseudo-Anosov maps, in particular explaining how the theory can be viewed from the perspective of outer automorphisms of surface groups. We will then present some recent work on the minimum dilatation problem with Eriko Hironaka, which shows a sharp lower bound for dilatations of fully-punctured pseudo-Anosov maps with at least two puncture orbits.

We construct a saturated system of 33 essential simple closed curves that are pairwise non-homotopic and intersect at most once on the oriented, closed surface of genus 3.

A spine for a group G acting properly discontinuously on a space E is a subset onto which there is a G-equivariant deformation retraction of E. For the space of lattices of covolume 1 in $\mathbb{R}^n$, the action of $SL_n(\mathbb{Z})$ admits a spine of minimal dimension called the well-rounded retract, consisting of the lattices whose shortest nonzero vectors span R^n. Whether an analogous spine of dimension $4g-5$ exists for the action of the mapping class group on the Teichmuller space of closed hyperbolic surfaces of genus g is an open problem. In a 1985 preprint, Thurston claimed to prove that the set $X_g$ of surfaces of genus g whose systoles (the shortest closed geodesics) fill (cut the surfaces into polygons) is a spine for the mapping class group. However, his argument had a serious gap. Whether or not $X_g$ is a spine, I will explain why its dimension is strictly larger than $4g-5$ in certain genera. The same construction shows that the set of surfaces whose systoles generate a finite-index subgroup in homology (a closer analogue of the well-rounded retract) does not contain any spine.

A countable Borel equivalence relation (CBER) $E$ on a Polish space $X$ is said to be treeable if there is a Borel forest $G\subseteq X$ whose trees are precisely the equivalence classes of said relation. $E$ is quasi-treeable if it has a Borel graphing, each of whose components is quasi-isometric to a tree. In joint work with Ruiyuan (Ronnie) Chen, Antoine Poulin and Anush Tserunyan, we show that quasi-treeable CBERs are treeable by giving a construction of a median graph associated to the quasi-treeing, which will be the main focus of this talk.

We prove that every action of a random group in the plain words density model on an n-dimensional CAT(0) cube complex has a global fix point with overwhelming probability. This generalizes previous work of Dahmani-Guirardel-Przytycki.

An action of a group on a space is called decent if every finitely generated subgroup all of whose elements have fixed-points has a global fixed-point. An example is the automorphism group of a tree or a finite product of trees. I will give a sufficient condition for a group acting on a restricted infinite product of trees to be decent. This allows to prove that every finitely generated subgroup of the Cremona group of $P^2$ all of whose elements are algebraic is bounded. Joint work with Anne Lonjou and Christian Urech.

The cross-sections of noncompact hyperbolic manifolds are flat manifolds of codimension 1. In 2002, Long and Reid proved that every flat manifold occurs as a cusp cross-section of some arithmetic hyperbolic manifold. I will discuss this result, and some number-theoretic obstructions to certain flat manifolds arising as cusps in certain classes of hyperbolic manifolds.

Regular trees of graphs are inverse limits of particularly simple inverse systems of finite graphs. They form a 1-dimensional subclass of the Markov compacta: a class of finitely describable inverse limits of simplicial complexes, which includes all boundaries of hyperbolic groups. I will discuss upcoming joint work with Jacek Swiatkowski in which we use Bowditch's canonical JSJ decomposition to characterize the 1-ended hyperbolic groups whose boundaries are (regular) trees of graphs.

Given a flat $n$-manifold, $M$, it is natural to ask which commensurability classes of hyperbolic $n+1$ manifold can contain $M$ as a cusp cross-section. Since commensurability classes of arithmetic hyperbolic manifolds can easily be described in terms of number-theoretic invariants they form a particularly accessible set of manifolds on which to explore such a question. I will explain why flat manifolds with $b_1\geq 3$ arise as cusp cross-sections in every commensurability class of arithmetic hyperbolic manifolds and how one can obstruct some flat manifolds with $b_1 < 3$ from arising in certain commensurability classes. This is ongoing joint work with Connor Sell.

In measured group theory (MGT), one studies group by their actions on finite or sigma-finite measure spaces. The notion of Measure Equivalence (ME), due to Gromov, is very similar to quasi-isometry and holds many powerful invariants. We will survey treeability in the ME context, look at the main obstruction to a strengthening of ME, namely orbit equivalence (OE). We will sketch why free groups of different rank are ME, but not OE. We will then look at these notions in the measure-class preserving context and see how cost is not useful here.

Hyperbolically embedded subgroups were defined by F. Dahmani, V. Guirardel and D. Osin as a generalization of peripheral structure of relatively hyperbolic groups. We revisit the definition of these subgroups using the Bowditch graph approach which was described by E. Martinez Pedroza and F. Rashid. Then we prove a combination theorem for hyperbolically embedded subgroups where each edge group of the splitting graph of groups is conjugate into a "subgroup" of a peripheral structure of the adjacent vertex group. Moreover, after defining groups with well-defined relative dehn function, we provide a similar combination theorem for these groups which follows from constructing a Cayley_Abel graph in the first part of this talk. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. This is a joint work with E. Martinez Pedroza.

Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. In their most general setting, very little is know about Artin groups. However many of the questions which are open for Artin groups can be easily answered for Artin monoids. This motivates the study of the connection between Artin groups and Artin monoids. In this talk, I will discuss two different geometric constructions including a CAT(0) cube complex and a version of the Cayley graph. These spaces illustrate the connection between the Artin monoid and group. I'll introduce some properties of these spaces how they might lead to further results.

The existence of a representation of the fundamental group of a 3-manifold with values in Homeo_+(S^1) potentially has a major impact on its topology via the L-space conjecture. We'll discuss how such representations arise from pseudo-Anosov flows and how an operation called stir-frying adds great variability to the construction. Time permitting, we'll discuss applications to the study of the L-space conjecture. This is joint work with Cameron Gordon and Ying Hu.