Abstracts

Gromov and Thurston constructed infinite families of manifolds that have metrics of pinched negative curvature but no metrics of constant negative curvature by taking cyclic branched covers of hyperbolic manifolds over codimension-two, totally geodesic submanifolds. I will discuss a result showing that some of these branched covers satisfy Singer's L2-Betti number vanishing conjecture via skew fields and special cube complex technology, partially answering a question raised by Gromov. Joint work with Boris Okun and Kevin Schreve.

A topological median structure on a topological space X is a continuous map m:X^3->X satisfying certain axioms. A CAT(0) cube complex X has a natural median structure, where m(a,b,c) is the unique point that belongs to three l^1-gedesics that connect each pair ab, ac, bc. In a work in progress, joint with Ken Bromberg and Michah Sageev, we show that median structures on R^n are locally induced by cubulations of neighborhoods. The proof is by induction on n, and requires us to prove the same theorem for ER homology manifolds.

Sam Shepherd and I proved that if two finite simplicial complexes have isomorphic universal covers and free fundamental groups, then they have a common finite cover. In the 1-dimensional case (graphs), this is Leighton's Theorem. I shall discuss various extensions of Leighton's Theorem and examples that illustrate its limitations, with a focus on coverings by quasitrees. I shall then outline a proof of our theorem, which involves the construction of auxiliary CAT(0) cube complexes.

Morse boundaries are designed to detect hyperbolic behavior in any geodesic metric space and have been shown to have many properties in common with hyperbolic boundaries.  Cannon and Thurston showed in the 1980’s that for M a hyperbolic 3-manifold that fibers over a circle, the inclusion of the fiber F into M induces a map of universal covers H2 → H3 that extends to a continuous surjection on their boundaries S1 → S2.  Later, this was generalized by Mitra (Mj) to the inclusion of a normal subgroup in a hyperbolic group and the induced map on their Gromov boundaries.  In this talk, I will discuss a project to investigate the existence and non-existence of Cannon-Thurston maps on Morse boundaries. (Joint work with Cordes, Goldborough, Sisto, and Zbinden.)

What are the possible discrete and Zariski-dense subgroups of SL(n,R) that contain a copy of SL(3,Z)? This question was posed by Madhav Nori in 1983. I will describe old results and new developements on that question. Based on joint work with Venkataramana.

A range of qualitative, geometric results are known about the cohomology of rank-1 local systems on quasi-projective varieties, and more is known in the special case of complements of complex hyperplane arrangements.  Even in that case, though, the general picture remains elusive.  The Schubert arrangements provide a nontrivial family of examples for which a complete and explicit description of the jump loci, in all cohomological degrees, is possible.  This is part of joint work with Prajwal Udanshive.

Given a group G and a homomorphism φ: G → ℤ, the Novikov ring Nov(G,φ) is a certain completion of the group ring ℤ[G] along the map φ. A Theorem of Sikorav states that ker(φ) is finitely generated if and only if the first homologies of G with coefficients in Nov(G,±φ) vanish.

In this talk, we will present a short exact sequence involving the Novikov rings Nov(G,±φ), the group ring of ker(φ), and the co-induced module of ker(φ), and use it to derive more properties of ker(φ), beyond those given Sikorav's Theorem and its generalisations. Applications to RFRS groups and to Poincaré-duality groups will be discussed. This is partly joint work with Dawid Kielak and Giovanni Italiano.

Burger-Mozes constructed examples of simple groups acting geometrically on a CAT(0) complex, which is a product of trees. As a counterpoint, we prove that every group acting geometrically on a CAT(0) cube complex which is not a product, admits a nontrivial quotient which also admits a geometric action on a CAT(0) cube complex. Our construction relies on the cubical version of small cancelation theory. This is joint work with M. Arenas and D. Wise.

I will show a very general construction of  characteristic classes of flat G-bundles for various groups G. These classes live in H^*(BG, W) with interesting coefficients construction of which is a part of the game. Then I will show some more delicate properties of these classes in special cases. Finally I will briefly discuss the outlook which includes relation to logic. The talk is mostly based on papers available on the arxiv. (Joint with Jan Dymara)

A Q-acyclic complex is a higher-dimensional analogue of a tree. We study a natural model of random 2-dimensional Q-acyclic complex, following enumerative work of Kalai and probabilistic work of Lyons. We are especially interested in its expected topological properties. We show that asymptotically almost surely, a random 2-dimensional Q-acyclic complex is aspherical. We also show that the torsion in homology grows exponentially fast in the number of faces, and the fundamental group is hyperbolic in the sense of Gromov. We will also discuss some open problems. In particular, the fundamental group of a random 2-dimensional Q-acyclic complex seems to be a candidate for a one-ended hyperbolic group without surface subgroups. This talk is based on joint work with Andrew Newman. 

We give a construction of uncountably many groups of type FP that does not involve Morse theory and is independent from the work of Bestvina-Brady, although there is some overlap between the groups constructed.  The main tool used is graphical small cancellation.  This is partially joint work with Tom Brown.  

A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or infinity. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and Hk(G,FG)=0 for k<n, then dim Hn(G,FG)=0,1 or ∞, significantly extending work of Farrell from 1975. Moreover, in the case dim Hn(G,FG)=1, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces.

In this talk, we talk about the tools needed to prove this result. We will introduce several coarse topological invariants of metric spaces, inspired by group cohomology. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group equipped with a proper left-invariant metric, then the coarse cohomological dimension of G coincides with its cohomological dimension whenever the latter is finite. Extending a result of Sauer, we show that coarse cohomological dimension is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one. 

All Coxeter groups are reflection groups.  And there are many contexts where "reflection group" and "Coxeter group" mean essentially the same thing.  In this talk I will highlight a case where these notions are different.   Concretely, a group generated by finitely many reflections preserving a non-singular quadratic form on a real metric vector space need not be a Coxeter group.

The Singer conjecture predicts that the $L^{2}$-Betti numbers $b^{(2)}_{k}(W_{L})$ of a RACG $W_{L}$ based on a flag triangulation $L$ of the $(n-1)$-sphere vanish, unless $k=n/2$.  A skew field interpretation of the $L^{2}$-Betti numbers shows that under some conditions this vanishing is preserved by edge subdivision/removal. This gives the following:

1. The Singer conjecture holds if $L$ is the barycentric subdivision of the boundary of a simplex,

2. The Singer conjecture holds if $L$ is the barycentric subdivision of a triangulation of an odd-dimensional sphere,

3. The only trivalent flag graph $L$ with non-zero $b^{(2)}_{2}(W_{L})$ is $K_{3,3}$. 

Based on joint work with Grigori Avramidi and Kevin Schreve.

Drilling a closed hyperbolic 3-manifold along an embedded geodesic is a crucial technique in low-dimensional topology, transforming the fundamental group of the manifold into a relatively hyperbolic group. In this talk, we extend this concept by proving that, under appropriate conditions, a similar "drilling" operation can be applied to a (Gromov) hyperbolic group with the 2-sphere boundary.

Our primary motivations and applications revolve around the Cannon Conjecture, which states that if the Gromov boundary of a hyperbolic group is homeomorphic to the 2-sphere, then the group is virtually (i.e., up to a finite-index subgroup) the fundamental group of a closed 3-manifold of constant negative curvature. We also explore the relatively hyperbolic counterpart—the Toral Relative Cannon Conjecture.

Using drilling, we show that if the Toral Relative Cannon Conjecture holds, then the Cannon Conjecture is valid for all residually finite hyperbolic groups. The Toral Relative Cannon Conjecture appears more accessible, owing to the presence of additional structure—abelian parabolic subgroups.

This is joint work with Daniel Groves, Peter Haïssinsky, Jason Manning, Alessandro Sisto, and Genevieve Walsh.

This is joint work with Yeeka Yau. I will explain why minimal elements of cone type components and Shi components form "Garside shadows" in Coxeter groups. This leads to convenient path systems in their Cayley graphs.

The separability of convex-cocompact subgroups of virtually special cubulated groups is a key part of the theory of special cube complexes, especially regarding the connection to 3-manifold theory. This was generalised to the separability of a product of two or three convex-cocompact subgroups, and in my recent work I generalise this further to a product of arbitrarily many such subgroups. In this talk I will discuss my result, along with relevant background regarding virtually special cubulated groups and also a more geometric version of my result in terms of elevations of routes to finite covers of special cube complexes.

A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of Röver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$.

As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups, the first is due to Caprace and Rémy. 

The polyhedral product is a functorial construction that assigns to each simplicial complex $K$ on $n$ vertices, and to each pair of topological spaces, $(X,A)$, a certain subspace, $\mathcal{Z}_K(X,A)$, of the cartesian product of $n$ copies of $X$. I will survey some of the connections between the duality properties of these spaces and the Cohen-Macaulay property of the original simplicial complex. I will also describe the resonance varieties of the polyhedral products $\mathcal{Z}_K(S^1,*)$, their propagation properties, and their scheme structure, leading to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. 

Ruth Charney, Corey Bregman and I constructed a space on which the outer automorphism group of a right-angled Artin group acts as symmetries, and proved that it is contractible with finite stabilizers.  More recently we showed that certain finite subgroups must have a fixed point in this space, and we are currently working on extending this to more general finite subgroups.  I will define the relevant spaces and give an update on our progress. 

One of Mike's most important contributions to topology was showing that aspherical manifolds have much more variety than had been anticipated.  His success with constucting examples with manifolds also called attention to our incapacity to construct any interesting Poincare duality groups that weren't manifolds.   This talk will have three parts: the first (joint with G.Lim) is an application of non-positively curved geometric constructions to the topology of positively curved spaces; the second (joint with J.Block) uses the same construction to construct nonexistent orbifolds and finally, if there's time, I'll explain some recent work (with S.Cappell and M.Yan) constructing over finite fields some manifolds which J.Fowler showed didn't exist over the rationals (which had, in turn, been spurred by a conjecture of Mike's).