
# Geometry and Topology Online

## Term II, 2021-22

Please contact Saul Schleimer or Robert Kropholler if you would like to speak or to suggest a speaker.

While this page is the seminar's "main page", I will attempt to also maintain an up-to-date listing at researchseminars.org.

The seminar will run weekly, with one 30 minute talk, followed by a 20 minute discussion/Q&A. The talk starts five minutes after the hour. We will open and close the Zoom session on the hour. Note that no password is required; links to the zoom session for each talk are below.

 Thursday January 13, 15:05 (UK time). Kim Ruane (Tufts) Torsion-free groups acting geometrically on the product of two trees Abstract: Given a group acting geometrically on product of two trees, we know that one visual boundary is the topological join of two Cantor sets. We prove that these groups are "boundary rigid": any $\CAT(0)$ space on which the group acts has visual boundary homeomorphic to such a join. Since there is no hyperbolicity going on here, one cannot expect that the natural equivariant quasi-isometry between an arbitrary $\CAT(0)$ space and the product of two trees to extend to any sort of map on boundaries, thus the proof requires new techniques. The proof uses work of Ricks on recognising product splittings from the Tits boundary as well as work of Guralnik and Swenson on general dynamics of a $\CAT(0)$ group on both the visual and Tits boundary. This is (recent) joint work with Jankiewicz, Karrer, and Sathaye.

 Thursday January 20, 15:05 (UK time). Vladimir Vankov (Southampton) Uncountably many quasi-isometric torsion-free groups Abstract: The study of quasi-isometries between finitely generated groups has traditionally been one of the more common questions of geometric group theory, which includes understanding the possible nature of quasi-isometry classes in general. There are several precedents for sets of uncountable cardinality to exhibit surprising behaviour differing from countable sets, especially when it comes to subgroups. We explore generalising constructions of uncountably many torsion groups falling into the same quasi-isometry class via commensurability, to the torsion-free setting. This is done by considering bounded cohomology and appealing to algebraic concepts classically found in finite group theory, in order to produce examples of a continuum of quasi-isometric and torsion-free, but pairwise non-isomorphic finitely generated groups.

 Thursday January 27, 15:05 (UK time). Davide Spriano (Oxford) Hyperbolic spaces for $\CAT(0)$ groups Abstract: $\CAT(0)$ spaces, as avatars of non-positive curvature, are both old and widely studied. Making up an important subclass are the $\CAT(0)$ cube complexes: spaces obtained by gluing Euclidean $n$-cubes along faces and satisfying an additional combinatorial conditions. Given such a space $X$, there are several techniques to construct associated spaces that "detect the hyperbolic behaviour" of $X$. All of these techniques rely on the combinatorial structure coming from the cubes. In this talk we will present a new approach to construct hyperbolic spaces on which $\CAT(0)$ groups act. We thus obtain characterisations of rank-one elements and recover rank-rigidity results. This is joint work with H. Petyt and A. Zalloum.

 Thursday February 3, 15:05 (UK time). None (None) None Abstract: None

 Thursday February 10, 15:05 (UK time). Macarena Arenas (Cambridge) Abstract: The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group $Q$, and producing as an output a hyperbolic group $G$ that maps onto $Q$ with finitely generated kernel. The "output group" $G$ is crafted by adding generators and relations to a presentation of $Q$, in such a way that these relations create enough "noise" in the presentation to ensure hyperbolicity. One can then lift pathological properties of $Q$ to (some subgroup of) $G$. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem. In this talk, I will explain Rips' result, mention some of its variations, and survey some tools and concepts related to these constructions, including small cancellation theory, cubulated groups, and asphericity. Time permitting, I will describe a variation of the Rips construction that produces cubulated hyperbolic groups of any desired cohomological dimension.

 Thursday February 17, 15:05 (UK time). Annette Karrer (Technion) Connected components of Morse boundaries of graphs of groups Abstract: Each finitely generated group has a topological space associated to it called the Morse boundary. This boundary generalizes the Gromov boundary of Gromov-hyperbolic groups and captures how similar the group is to a Gromov-hyperbolic group. In this talk, we will study connected components of Morse boundaries of a graph of groups $G$. We will focus on the case where the edge groups are undistorted and do not contribute to the Morse boundary of $G$. We will describe the connected components of the Morse boundary of $G$ using the associated Bass-Serre tree. We will see that every connected component of the Morse boundary with at least two points originates from the Morse boundary of a vertex group. This is joint work with Elia Fioravanti.

 Thursday February 24, 15:05 (UK time). Luke Jeffreys (Bristol) Non-planarity of $\SL(2,\ZZ)$-orbits of origamis in genus two Abstract: Origamis (also known as square-tiled surfaces) arise naturally in a variety of settings in low-dimensional topology. They can be thought of as surfaces obtained by gluing the sides of a collection of unit squares. As such, they generalise the torus which can be obtained by gluing the sides of a single square. An origami is said to be primitive if it is not a cover of a lower genus origami. In this talk, I will describe how one can define an action of the matrix group $\SL(2,\ZZ)$ on primitive origamis. In genus two (with one singularity), the orbits of this action were classified by Hubert-Lelièvre and McMullen. By considering a generating set of size two for $\SL(2,\ZZ)$, we can turn these orbits into an infinite family of four-valent graphs. For a specific generating set, I will explain how all but two of these graphs are non-planar. I will also discuss why this gives indirect evidence for McMullen's conjecture that these graphs form a family of expanders. This is joint work with Carlos Matheus.

 Thursday March 3, 15:05 (UK time). Armando Martino (Southampton) On automorphisms of free groups and nearly canonical trees Abstract: I will discuss some open problems for automorphisms of free groups; whether centralisers are finitely generated, whether their mapping tori have well-behaved automorphism group, and whether the conjugacy problem is solvable. I will explain some new partial results, using techniques involving canonical trees. This is joint work Naomi Andrew, and various others.

 Thursday March 10, 15:05 (UK time). Sam Hughes (Oxford) Irreducible lattices fibring over the circle Abstract: Let $n \geq 2$ and let $\Lambda$ be a lattice in a product of simple non-compact Lie groups with finite centre. An application of the Margulis normal subgroup theorem implies that if $H^1(\Lambda)$ is non-zero, then $\Gamma$ is reducible. In the more general $\mathrm{CAT}(0)$ setting there are many irreducible lattices with non-vanishing first cohomology. In this case we can deploy the BNSR invariants and investigate how far these cohomology classes are from a fibration of finite type CW complexes. In this talk we will combine the groups of Leary and Minasyan with the technology of Bestvina and Brady to construct the first examples of irreducible lattices which fibre over the circle.

 Thursday March 17, 15:05 (UK time). Rylee Lyman (Rutgers) Folding-like techniques for $\CAT(0)$ cube complexes Abstract: In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on $\CAT(0)$ cube complexes. We extend their techniques to fundamental groups of non-positively curved cube complexes.

Information on past talks. This line was last edited on Thu 17 Mar 2022 14:04:33 GMT