# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term III, 2017-2018

 Thursday April 26, 15:00, room MS.04 Pierre-Emmanuel Caprace (Louvain) Almost simplicity of commensurators of free and surface groups. Abstract: The group $\Aut(F_n)$ is a prominent character in geometric group theory. The goal of this talk is to advertise a larger group, namely the group of abstract commensurators of $F_n$, denoted by $\Comm(F_n)$. Bartholdi and Bogopolski have shown that $\Comm(F_n)$ is infinitely generated. A. Lubotzky has asked whether $\Comm(F_n)$ is simple. I will explain that $\Comm(F_n)$ is almost simple. The relative commensurator of $F_n$ in the automorphim group of its Cayley tree, and commensurator groups of surface groups, will also be mentioned.

 Thursday May 3, 15:00, room MS.04 Jim Howie (Heriot-Watt) Two-complexes with the non-positive immersion property Abstract: A two-complex $X$ has non-positive immersions if every compact connected, non-contraactible two-complex $Y$ admitting an immersion $Y \to X$ has non-positive Euler characteristic. This concept was introduced by Dani Wise in connection with Baumslag's conjecture that one-relator groups are coherent. A theorem of Helfer and Wise - and independently of Louder and Wilton - says that the geometric realisation of a torsion-free one-relator group presentstion has the non-positive immersions property. I will present a relative version of this theorem, together with a number of related results. This is joint work with Hamish Short.

 Thursday May 10, 15:00, room MS.04 Ruth Charney (Brandeis) Quasi-mobius maps of Morse boundaries Abstract: Boundaries of hyperbolic groups have played an important role in low-dimensional topology and geometric group theory. By restricting to rays satisfying the Morse property, one can define an analogous boundary for more general groups. Inspired by a theorem of Paulin, we give precise conditions for when a homeomorphism between the Morse boundaries of two groups is induced by a quasi-isometry of the groups themselves. (Joint work with M. Cordes and D. Murray.)

 Thursday May 17, 15:00, room MS.04 Luis Paris (Bourgogne) Commensurability in Artin groups Abstract: Recall that an Artin group is a group defined by a presentation with relations of the form $sts\cdots = tst\cdots$ where the the word on the left hand side and the word on the right hand side having the same length. These groups were introduced by Tits in the 1960s and they are involved in several fields such as the study of singularities, low-dimensional topology, or geometric group theory. There are very few results valid for all Artin groups and the theory mainly consists on the study of particular families. The most studied and better understood are the family of right-angled Artin groups (RAAG) and the family of spherical type Artin groups. The latter is the topic of the talk. Following a result from 2004 where I proved that two spherical type Artin groups are isomorphic if and only if they have the same presentation, the project of classifying these groups up to commensurability was born. This relation is a kind of equality up to finite index. Our aim will be to give a general presentation on this question and to explain recent advances. This is a joint work with Maria Cumplido Cabello

 Thursday May 17, 16:00, room MS.04 Vaibhav Gadre (Glasgow) Vanishing of the Lyapunov expansion exponent for non-uniform lattices in $\SL(2,\RR)$ Abstract: For a finitely generated group of circle diffeomorphisms, Deroin-Kleptsyn-Navas defined a Lyapunov expansion exponent at a point on the circle. With J. Maher and G. Tiozzo we prove that for a non-uniform lattice in $\SL(2,\RR)$, the exponent vanishes at Lebesgue almost every point of the circle. This answers a question of Deroin-Kleptsyn-Navas. The key tool is the statistics of word length along random geodesics. If time permits, I will explain the analogous statistical results along random Teichmuller geodesics.

 Thursday May 24, 15:00, room MS.04 TBA (TBA) TBA Abstract: TBA

 Thursday May 31, 15:00, room MS.04 TBA (TBA) TBA Abstract: TBA

 Thursday June 7, 15:00, room MS.04 Richard Hind (Notre Dame) Embedding and packing Lagrangian tori Abstract: Embedding and packing problems are well studied in symplectic topology. The maximal size of a ball which can be embedded in a symplectic manifold is its Gromov width, and its packing numbers give the number of balls of a fixed size which admit disjoint embeddings. It also makes sense to talk about embedding and packing Lagrangian torus submanifolds, even though there is no longer a volume obstruction as in the ball case. We will obtain optimal bounds for Lagrangian tori in the 4-ball, but natural packings coming from toric pictures turn out not to be maximal. This is joint work with Ely Kerman and Emmanuel Opshtein.

 Monday June 11, 18:15, room MS.02 Jeffrey Weeks (Independent) The shape of space Abstract: When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a “multiconnected universe”. Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how satellite data provide tantalizing clues to the true shape of our universe. The only prerequisites for this talk are curiosity and imagination. For middle school and high school students, Warick students and faculty, people interested in astronomy, and all members of the Coventry community.

 Tuesday June 12, 15:00, room B3.03 Jeffrey Weeks (Independent) Visualising four dimensions Abstract: This talk will introduce a method for learning to visualize four-dimensional space, give participants a chance to work on some 4D visualization exercises in small groups, and then present a few solutions using interactive 4D graphics software. The exercises range from elementary to advanced, so everyone from first-year undergraduates to seasoned geometers should find something they like.

 Thursday June 21, 15:00, room MS.04 TBA (TBA) TBA Abstract: TBA

 Thursday June 28, 10:00, room MS.04 Luigi Caputi (Regensburg) A coarse version of Hochschild and cyclic homology Abstract: Bornological coarse spaces are generalizations of metric spaces. Homological invariants of such spaces are given by coarse homology theories, which are functors from the category of bornological coarse spaces to any stable cocomplete $\infty$-category, satisfying additional axioms. Among the main examples we have a coarse version of ordinary homology, topological and algebraic $K$-theory. In the talk we define equivariant coarse versions of the classical Hochschild and cyclic homologies. If $k$ is a field of coefficients, the evaluation at the one point space induces equivalences with the classical Hochschild and cyclic homologies of $k$. In the equivariant setting, the $G$-equivariant coarse Hochschild (cyclic) homology of the discrete group $G$ agrees with the classical Hochschild (cyclic) homology of the associated group algebra $k[G]$. Moreover, the forget-control map for coarse Hochschild homology agrees with the generalized assembly map for Hochschild homology, when one considers the family of finite subgroups.

Information on past talks. This line was last edited Monday, October 2, 2017 02:30:50 PM BST