
# Geometry and Topology Seminar

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

 Thursday October 5, 15:00, room MS.03 None (None) None Abstract: None

 Thursday October 12, 15:00, room MS.03 None (None) None Abstract: None

 Thursday October 19, 15:00, room MS.03 John Jones (Warwick) Homotopy equivalences of configuration spaces and the Grothendieck - Teichmüller Lie algebra Abstract: The Grothendieck - Teichmüller Lie algebra $\mathfrak{grt}$ is a Lie algebra, over the rational numbers $\QQ$, which is clearly very interesting and equally clearly not very well-understood. It crops up in many different areas of mathematics. In this talk I will explain how the Lie algebra $\mathfrak{grt}$ is related to the space of homotopy equivalences of the configuration spaces $F(k, \RR^n)$ of $k$ distinct ordered points in $\RR^n$. The talk will very down to earth and much of it will be devoted to explaining, in as concrete a way as possible, the background needed for the statements of the results.

 Thursday October 26, 15:00, room MS.03 Richard Webb (Cambridge) Searching for $\CAT(0)$ complexes, and where not to look Abstract: In their seminal work Masur and Minsky state that it is an interesting question whether the arc-and-curve complex admits a $\CAT(0)$ metric. We will start by motivating this question further, and in particular, explaining the desire to go beyond already known $\CAT(0)$ spaces e.g. (the completion of) the Weil--Petersson metric on Teichmueller space. We will then discuss where not to look for $\CAT(0)$ spaces; we will show that the majority of arc complexes, and, all but finitely many disc complexes of handlebodies and free splitting complexes do not admit $\CAT(0)$ metrics with finitely many/tractable shapes. Furthermore a possible strategy for dealing with the arc-and-curve complex will be discussed.

 CANCELLED Thursday November 2, 15:00, room MS.03 Michael Shapiro (Bath) The Heisenberg group has rational growth in all generating sets Abstract: Given a group $G$ and a finite generating set $\calG$ the (spherical) growth function $f_\calG(x) = a_0 + a_1 x + a_2 x^2 + \ldots$ is the series whose coefficients $a_n$ count the number of group elements at distance $n$ from the identity in the Cayley graph $\Gamma_\calG(G)$. For hyperbolic groups and virtually abelian groups, this is always the series of a rational function regardless of generating set. Many other groups are known to have rational growth in particular generating sets. In joint work with Moon Duchin, we show that the Heisenberg group also has rational growth in all generating sets. The first ingredient in this result is to compare the group metric, which we can see as a metric on the integer Heisenberg group with a metric on the real Heisenberg group. This latter is induced by a norm in the plane which is in turn induced by a projection of the generating set. The second ingredient is a wondrous theorem of Max Bensen regarding summing the values of polynomical over sets of lattice points in families of polytopes. We are able to bring these two ingredients together by showing that every group element has a geodesic whose projection into the plane fellow-travels a well-behaved set of polygonal paths.

 Thursday November 9, 15:00, room MS.03 Michael Magee (Durham) Word measures on unitary groups Abstract: This is joint work with Doron Puder (Tel Aviv University). For a positive integer $r$, fix a word $w$ in the free group on $r$ generators. Let $G$ be any group. The word $w$ gives `word map' from $G^r$ to $G$: we replace the generators in $w$ by the corresponding elements of $G$. We again call this map $w$. The push-forward of Haar measure under $w$ is called the $w$-measure on $G$. We are interested in the case $G = U(n)$, the compact Lie group of $n$-dimensional unitary matrices. A motivating question is: to what extent do the $w$-measures on $U(n)$ determine algebraic properties of the word $w$? We proved in our first paper that one can detect the 'stable commutator length' of $w$ from the $w$-measures on $U(n)$. Our main tool was a formula for the Fourier coefficients of $w$-measures; the coefficients are rational functions of the dimension $n$, for reasons coming from representation theory. We can now explain all the Laurent coefficients of these rational functions in purely topological terms. I'll explain all this in my talk, which should be broadly accessible and of general interest. I'll also invite the audience to consider some remaining open questions.

 Thursday November 16, 15:00, room MS.03 Christoforos Neofytidis (Geneva) Anosov tori in three-manifolds Abstract: We study the effect of the mapping class group of a reducible three-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. In particular, we show that a closed oriented reducible three-manifold admits an Anosov torus if and only if one of its prime summands admits an Anosov torus, answering a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures. This is joint work with Shicheng Wang.

 Thursday November 23, 15:00, room MS.03 Selim Ghazouani (Warwick) The complex hyperbolic geometry of certain moduli spaces of tori Abstract: Generalising an idea of Thurston, Veech defines homogeneous structures on several moduli spaces of flat surfaces with cone type singularities. The specific case of tori provides natural (non-complete) complex hyperbolic structures on certain complex manifolds. We provide an interpretation of the metric completion of these manifolds in terms of degenerations of the underlying flat structures. This leads to a construction of complex hyperbolic cone-manifolds of finite volume, whose holonomy are in a finite number of case an arithmetic lattice. This is joint work with Luc Pirio (CNRS Versailles).

 Thursday November 30, 15:00, room MS.03 Kate Vokes (Warwick) Geometry of the separating curve graph Abstract: There are many graphs and complexes we can associate to a surface whose vertices are curves or collections of curves in the surface. A first example is the curve graph, which has a vertex for each isotopy class of curves, with edges corresponding to disjointness. These complexes have proved to be useful tools in studying mapping class groups and Teichmüller spaces. Masur and Minsky, in 2000, gave a distance estimate for the mapping class group in terms of a sum of certain projections to the curve graphs of subsurfaces. We will present a result which gives a similar distance estimate for the separating curve graph, making use of the concept of hierarchical hyperbolicity defined by Behrstock, Hagen and Sisto.

 Thursday December 7, 15:00, room MS.03 Ric Wade (Oxford) Relative automorphisms of right-angled Artin groups Abstract: We look at the group of outer automorphisms of a right-angled Artin group preserving a set of special subgroups (the subgroups coming from subgraphs of the defining graph). I will give some examples to show how these groups arise naturally in the study of the whole automorphism group, and sketch how such groups are finitely generated, and up to finite index, have finite classifying spaces. This is joint work with Matthew B Day.

Information on past talks. This line was last edited Monday, October 2, 2017 12:51:22 PM BST.