Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will present a new proof of this using groupoids, and then talk about two generalisations of the theorem that can also be tackled with this groupoid approach: one gives us control over the local structure of the common finite cover, and the other deals with graphs of spaces.

# Past Junior Topology and Group Theory Seminar

The last decade or so has seen substantial progress in the theory of (outer) automorphism groups of right-angled Artin groups (RAAGs), spearheaded by work of Charney and Vogtmann. Many of the techniques used for RAAGs also apply to a wider class of groups, graph products of finitely generated abelian groups, which includes right-angled Coxeter groups (RACGs). In this talk, I will give an introduction to automorphism groups of such graph products, and describe recent developments surrounding the outer automorphism groups of RACGs, explaining the links to what we know in the RAAG case.

It is often fruitful to study an infinite discrete group via its finite quotients. For this reason, conditions that guarantee many finite quotients can be useful. One such notion is residual finiteness.

A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

The discrete fundamental groups of a metric space can be thought of as fundamental groups that `ignore' closed loops up to some specified size R. As the parameter R grows, these groups have been used to produce interesting invariants of coarse geometry. On the other hand, as R gets smaller one would expect to retrieve the usual fundamental group as a limit. In this talk I will try to briefly illustrate both these aspects.

The Poincare Conjecture was first formulated over a century ago and states that there is only one closed simply connected 3-manifold, hinting at a link between 3-manifolds and their fundamental groups. This seemingly basic fact went unproven until the early 2000s when Perelman proved Thurston's much more powerful Geometrisation Conjecture, providing us with a powerful structure theorem for understanding all closed 3-manifolds.

In this talk I will introduce the results developed throughout the 20th century that lead to Thurston and Perelman's work. Then, using Geometrisation as a black box, I will present a proof of the Poincare Conjecture. Throughout we shall follow the crucial role that the fundamental group plays and hopefully demonstrate the geometric and group theoretical nature of much of the modern study of 3-manifolds.

As the title suggests, no prior understanding of 3-manifolds will be expected.

Buildings are geometric objects, originally introduced by Tits to study Lie groups that act on their corresponding building. Apart from their significance for Lie groups, buidings and their automorphism groups are a rich source of examples for groups with interesting properties (for example, it is a result of Caprace that some buildings admit an automorphism group which is compactly generated, abstractly simple and locally compact). Right Angled Buildings (RABs) are a specific kind of building whose geometry can be well understood as it resembles the geometry of a tree. This allows one to generalise ideas like the Burger-Mozes universal groups to the setting of RABs.

I plan to give an introduction to RABs. As a complete formal introduction to buildings would take more than an hour, I will instead present various illustrative examples to give you an idea of what you should have in mind when you think of a (right-angled) building. I will be as formal as I can in presenting the basic features of buildings - Coxeter complexes, chambers, apartments, retractions and residues. In the remaining time I will say as much as I can about the geometry of RABs, and explain how to use this geometry to derive a structure theorem for the automorphism group of a RAB, towards a definition of Burger-Mozes universal groups for RABs.

Let $G$ be a group which splits as $G = F_n * G_1 *...*G_k$, where every $G_i$ is freely indecomposable and not isomorphic to the group of integers. Guirardel and Levitt generalised the Culler- Vogtmann Outer space of a free group by introducing an Outer space for $G$ as above, on which $\text{Out}(G)$ acts by isometries. Francaviglia and Martino introduced the Lipschitz metric for the Culler- Vogtmann space and later for the general Outer space. In a joint paper with Francaviglia and Martino, we prove that the group of isometries of the Outer space corresponding to $G$ , with respect to the Lipschitz metric, is exactly $\text{Out}(G)$. In this talk, we will describe the construction of the general Outer space and the corresponding Lipschitz metric in order to present the result about the isometries.

In geometric group theory we study groups by their actions on metric spaces. Although a given group might admit many actions on different metric spaces, on a large scale these spaces will all look similar, and so the large scale properties of a space on which a group acts are intrinsic to the group. One particularly natural example of a large scale property used in this way is the Gromov boundary of a hyperbolic metric space. This is a topological space that can be thought of as compactifying the metric space at infinity.

In this talk I will describe some constructions of spaces occurring in this way with nasty, fractal-like properties. On the other hand, there are limits to how pathological these spaces can be: theorems of Bestvina and Mess, Bowditch and Swarup imply that boundaries of hyperbolic groups are locally path connected whenever they are connected. I will discuss these results and some generalisations.

A stacking is a lift of an immersion of graphs $A\to B$ to an embedding of $A$ into the product of $B$ with the real line; their existence relates to orderability properties of groups. I will describe how Louder and Wilton used them to prove Wise's "$w$-cycles" conjecture: given a primitive word $w$ in a free group $F$, and a subgroup $H < F$, the number of conjugates of $H$ which intersect $<w>$ nontrivially is at most rank($H$). I will also discuss applications of the result to questions of coherence, and possible extensions of it.

I will give a self-contained introduction to the theory of cross ratios on boundaries of Gromov hyperbolic and CAT(-1) spaces, focussing on the connections to the following two questions. When are two spaces with the 'same' Gromov boundary isometric/quasi-isometric? Are closed Riemannian manifolds completely determined (up to isometry) by the lengths of their closed geodesics?