# Geometry and Topology Online

## Term III, 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 be hybrid, and will be 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 April 28, 15:05 (UK time). Zoom meeting Marco Linton (Warwick) Hyperbolicity of certain one-relator groups Abstract: The primitivity rank of an element $w$ of a free group $F$ is defined as the minimal rank of a subgroup containing w as an imprimitive element. Recent work of Louder and Wilton has shown that there is a strong connection between this quantity and the subgroup structure of the one-relator group $F/\langle \langle w \rangle \rangle$. In particular, they show that one-relator groups whose defining relation has primitivity rank at least three cannot contain Baumslag—Solitar subgroups, leading them to conjecture that such groups are hyperbolic. In this talk, I will confirm and strengthen this conjecture, providing some applications.

 Thursday May 5, 15:05 (UK time). Marco Barberis (Warwick) Curve graphs: exhaustions by rigid sets and the co-Hopfian property Abstract: Since Ivanov's celebrated first result, many rigidity theorems for various variants of the curve graph of surfaces have been proven. Among these, there is a cluster of results regarding the existence of exhaustion via finite subgraphs which are rigid (that is such that every embedding is induced by an automorphism of the whole graph). From this property, interesting per se, the co-Hopfian property of the graphs immediately follows. In this talk I will present the classical results in the fields, as well as some new cases, which point toward conjecturing that most curve graphs on finite-type surfaces should admit exhaustions by rigid sets, in line with Ivanov's Metaconjecture.

 Thursday May 12, 15:05 (UK time). Canonical forms for free group automorphisms Abstract: The Nielsen-–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan canonical form. I will discuss my progress in developing a similar decomposition for free group automorphisms. (Un)fortunately, free group automorphisms can have arbitrarily complicated behaviour. This forms a significant barrier to translating specific arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

 Thursday May 19, 15:05 (UK time). Susan Hermiller (Nebraska) Formal conjugacy growth for graph products Abstract: The conjugacy growth series of a finitely generated group measures the growth of conjugacy classes, in analogy with the standard growth series that measures the growth of elements of the group. In contrast, though, conjugacy growth series are rarely rational, and even for free groups with standard generating sets, the series are transcendental and their formulas are rather complicated. In this talk I will discuss several results on conjugacy growth and languages in graph products, including a recursive formula for computing the conjugacy growth series of a graph product in terms of the conjugacy growth and standard growth series of subgraph products. In the special case of right-angled Artin groups I will also discuss a another formula for the conjugacy growth series based on a natural language of conjugacy representatives. This is joint work with Laura Ciobanu and Valentin Mercier.

 Thursday May 26, 15:05 (UK time). Jone Lopez de Gamiz (Warwick) On finitely generated normal subgroups of right-angled Artin groups Abstract: In general, subgroups of RAAGs are known to have wild structure and bad algorithmic behaviour. However, in this talk we will see that finitely generated normal subgroups are much more tame. More precisely, we will show that a finitely generated normal subgroup of a RAAG is virtually co-abelian. We will then discuss some algorithmic consequences, such as the decidability of the conjugacy and the membership problems. We will finally discuss residual properties, such as conjugacy separability, for finitely generated normal subgroups of RAAGs.

 Thursday June 2, 15:05 (UK time). None (None) None Abstract: None

 Thursday June 9, 15:05 (UK time). None (None) None Abstract: None

 Thursday June 16, 15:05 (UK time). Naomi Andrew (Southampton) Baumslag-Solitar groups, automorphisms and generalisations Abstract: Baumslag-Solitar groups are a well known family in geometric group theory, providing useful (counter)examples - such as groups that are Hopfian but not residually finite. Recently, Ian Leary and Ashot Minasyan introduced a generalisation, finding even more counterexamples - notably groups that are $\CAT(0)$ but not biautomatic. Outer automorphism groups of Baumslag-Solitar groups range from finite to not even finitely generated, with proofs (and re-proofs) across several authors and years. In this talk I will summarise (some) of what is known about the automorphisms of Baumslag-Solitar groups, and the more modern, Bass-Serre theoretic techniques that can be used to prove them. I'll then discuss my work with Sam Hughes to extend these results to the automorphisms of Leary-Minasyan groups.

 Thursday June 23, 15:05 (UK time). Lorenzo Ruffoni (Tufts) Strict hypbolisation and special cubulation Abstract: Gromov introduced some "hyperbolisation" procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined "strict hyperbolisation" procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct new examples of manifolds and groups with negative curvature, and other prescribed features. We construct actions of the resulting groups on $\CAT(0)$ cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite. This is joint work with J. Lafont.

 Thursday June 30, 15:05 (UK time). Gareth Wilkes (Cambridge) Residual properties of graphs of $p$-groups Abstract: When groups may be built up as graphs of 'simpler' groups, it is often of interest to study how good residual finiteness properties of the simpler groups can imply residual properties of the whole. The essential case of this theory is the study of residual properties of finite groups. In this talk I will discuss the question of when a graph of finite $p$-groups is residually $p$-finite, for $p$ a prime. I will describe the previous theorems in this area for one-edge and finite graphs of groups, and their method of proof. I will then state a generalisation of these theorems to potentially infinite graphs of groups, together with an alternative and perhaps more natural method of proof. Finally I will briefly describe a usage of these results in the study of accessibility—namely the existence of a finitely generated inaccessible group which is residually $p$-finite.

Information on past talks. This line was last edited on Thu 14 Apr 2022 15:24:54 BST

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