The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered.
In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by discussing ongoing work in the area.
This talk contains some joint work with C.Abbott, D.Osin and A.Rasmussen.[-]

The theory of group actions on CAT(0) cube complexes has exerted a strong influence on geometric group theory and low-dimensional topology in the last two decades. Indeed, knowing that a group G acts properly and cocompactly on a CAT(0) cube complex reveals a lot of its algebraic structure. However, in general, "cubulations'' are non-canonical and the group G can act on cube complexes in many different ways. It is thus natural to try and formulate a good notion of "space of all cubulations of G'', which would prove useful in the study of Out(G) for quite general groups G. I will describe some results in this direction, based on joint works with J. Beyrer and M. Hagen.[-]

Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), cocompactly cubulated groups, hyperbolic groups, Coxeter groups and the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. I will discuss some of these examples as well as some general constructions and properties of shortcut graphs and groups.[-]

A well-known result of Davis-Januszkiewicz is that every right-angled Artin group (RAAG) is commensurable to some rightangled Coxeter group (RACG). In this talk we consider the converse question: which RACGs are commensurable to some RAAG? To do so, we investigate some natural candidate RAAG subgroups of RACGs and characterize when such subgroups are indeed RAAGs. As an application, we show that a 2-dimensional, one-ended RACG with planar defining graph is quasiisometric to a RAAG if and only if it is commensurable to a RAAG. This talk is based on work joint with Pallavi Dani.[-]

If a group G acts isometrically on a metric space X and Y is any metric space that is quasi-isometric to X, then G quasi-acts on Y. A fundamental problem in geometric group theory is to straighten (or quasi-conjugate) a quasi-action to an isometric action on a nice space. We will introduce and investigate discretisable spaces, those for which every cobounded quasi-action can be quasi-conjugated to an isometric action of a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups have this property, but it holds much more generally. For instance, we show that every hyperbolic group is either commensurable to a cocompact lattice in rank one Lie group, or it is discretisable.
We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.[-]

Parabolic subgroups are the fundamental building blocks of Artin groups. These subgroups are isomorphic copies of smaller Artin groups nested inside a given Artin group. In this talk, I will discuss questions surrounding how parabolic subgroups sit inside Artin groups and how they interact with each other. I will show that, in an FC type Artin group, the intersection of two finite type parabolic subgroups is a parabolic subgroup. I will also discuss how parabolic subgroups might be used to construct a simplicial complex for Artin groups similar to the curve complex for mapping class groups. This talk will focus on using geometric techniques to generalize results known for finite type Artin groups to Artin groups of FC type.[-]

The famous Hanna Neumann Conjecture (now the Friedman--Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to 'quantify' this bound with respect to the rank of the join of H and K, the subgroup generated by H and K. In this talk I describe what is known about the set of realizable values (rank of join, rank of intersection) for arbitrary H, K, and about my recent results in this direction. In particular, we resolve the remaining open case (m=4) of Guzman's `Group-Theoretic Conjecture' in the affirmative. This has some interesting corollaries for the geometry of hyperbolic 3-manifolds. Our methods rely on recasting the topological pushout of core graphs in terms of the Dicks graphs introduced in the context of his Amalgamated Graph Conjecture. This allows to translate the question of existence of a pair of subgroups H,K with prescribed ranks of joins and intersections into graph theoretic language, and completely resolve it in some cases. In particular, we completely describe the locus of realizable values of ranks in the case when the rank of one of the subgroups H,K equals two.[-]