
# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term II, 2019-2020

 Thursday January 9, 15:00, room MS.03 Henry Segerman (Oklahoma State) Essential loops in taut ideal triangulations Abstract: We combinatorialise a technique of Novikov. We use this to prove that, in a three-manifold equipped with a taut ideal triangulation, any vertical or normal loop is essential in the fundamental group. This is joint work with Saul Schleimer.

 Thursday January 16, 15:00, room MS.03 None (None) None Abstract: None

 Thursday January 23, 15:00, room MS.03 Jason Manning (Cornell) Groups acting improperly on cube complexes Abstract: A (proper) cubulation of a group $G$ is a $\CAT(0)$ cube complex $X$ together with a proper action of $G$ on $X$. Partial cubulation drops the properness assumption. Sometimes a partial cubulation can be promoted to a proper cubulation. Sometimes a proper cubulation can be improved by passing through a partial cubulation. I'll give examples of theorems of both types, focusing on hyperbolic and relatively hyperbolic groups. This is joint work with Daniel Groves.

 Thursday January 30, 15:00, room MS.03 Nima Hoda (ENS Paris) Shortcut graphs and groups Abstract: Shortcut graphs are graphs in which sufficiently long 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 metric graph theory and geometric group theory. Among the these groups we find: 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 actually satisfy an even stronger form of the shortcut property. I will discuss the general constructions and properties of shortcut graphs and groups and illustrate the definitions with several of the above examples.

 Thursday February 6, 14:00, room MS.01 Romain Tessera (Jussieu-Paris) Quantitative measure equivalence Abstract: Measure equivalence is an equivalence relation between countable groups that has been introduced by Gromov. A fundamental instance are lattices in a same locally compact group. According to a famous result of Orstein Weiss, all countable amenable groups are measure equivalent, meaning that geometry is completely rubbed out by this equivalence relation. Recently some more restrictive notions have been introduced such as $L^p$-measure equivalence, where the associated cocycles are assumed to be $L^p$-integrable. By contrast, a lot of surprising rigidity results have been proved: for instance Bowen has shown that the volume growth is invariant under $L^1$-measure equivalence, and Austin proved that nilpotent groups that are $L^1$-measure equivalent have bi-Lipschitz asymptotic cones. In this work we extend this study by trying to understand more systematically how the geometry survives through measure equivalence when an integrability condition is imposed. Our results go in two directions: we prove rigidity results, culminating for amenable groups with a general monotonicity result for the isoperimetric profile, and flexibility results showing that in many instances, the previous result is close to being optimal. We also prove a rigidity result for hyperbolic groups, showing the optimality of a result of Shalom for lattices in $\SO(n,1)$.

 Thursday February 6, 15:00, room MS.03 Radhika Gupta (Bristol) Non-uniquely ergodic arational trees in the boundary of Outer space Abstract: The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arational trees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called non-uniquely ergodic. In this talk, I will first show a construction of such laminations for surfaces due to Gabai. I will then present a construction of a similar phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.

 Thursday February 13, 15:00, room MS.03 Rachel Roberts (Washington) Taut foliations from double-diamond replacements Abstract: I will give some basic background about taut foliations and their relationship to Heegaard-Floer homology, and define a readily recognizable type of sutured manifold decomposition, which we call double-diamond taut. I will show that if there is a double-diamond taut sutured manifold decomposition, then every boundary slope except one is strongly realized by a co-oriented taut foliation; that is, the foliation intersects the boundary of M transversely in a foliation by curves of that slope. This work is joint with Charles Delman.

 Thursday February 20, 15:00, room MS.03 Eric Babson (Davis) Random fundamental groups Abstract: Various models of random two dimensional complexes have a sharp transition to simple connectivity at a density different from that for vanishing homology providing a range in which the fundamental group is nontrivial, Kazhdan and hyperbolic with no nontrivial small quotients. Along the way to apply Gromov's local to global hyperbolicity criterion we study two dimensional simplicial complexes with few faces and find that these have a form of positive curvature resulting in very restricted topology. One case of this phenomenon in two dimensions remains open and there are guesses for higher dimensional analogs. This project is joint work with Hoffman and Kahle.

 Thursday February 27, 15:00, room MS.03 Tim Riley (Cornell) Conjugator length Abstract: The conjugator length function of a finitely generated group $G$ maps a natural number $n$ to the minimal $N$ such that if $u$ and $v$ are words representing conjugate elements of $G$ with the sum of their lengths at most $n$, then there is a word $w$ of length at most $N$ such that $u w = w v$ in $G$. I will explore why this function is important, will describe some recent results with Martin Bridson and Andrew Sale on how it can behave, and will highlight some of the many open questions about conjugator length.

 Thursday March 5, 15:00, room MS.03 TBA (TBA) TBA Abstract: TBA

 Thursday March 12, 15:00, room MS.03 CANCELLED Talia Fernós (UNC) Boundaries and $\CAT(0)$ cube complexes Abstract: The universe of $\CAT(0)$ cube complexes is rich and diverse thanks to the ease by which they can be constructed and the many of natural metrics they admit. As a consequence, there are several associated boundaries, such as the visual boundary and the Roller boundary. In this talk we will discuss some relationships between these boundaries, together with the Furstenberg-Poisson boundary of a "nicely'' acting group.

Information on past talks. This line was last edited 2019-01-11.