Geometry and Topology Seminar

Thursday October 2, 15:00, room MS.04

John Jones (Warwick)

Closed geodesics and string homology

Abstract: (Joint work with John McCleary.) The study of closed geodesics on a Riemannian manifold $M$ is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse--Bott theory to give a topological condition on the free loop space of a closed (compact with no boundary) manifold $M$ which ensures that any Riemannian metric on $M$ has an infinite number of closed geodesics. This gives a very close connection between closed geodesics and the topology of loop spaces. Nowadays it is known that there is a rich algebraic structure associated to the topology of spaces loop -- this is the theory of string homology initiated by Chas and Sullivan in 1999.

In our work, we use the ideas of string homology to give new results on the existence of an infinite number of closed geodesics. I will explain some of the background of what has come to be known as the closed geodesics problem and some of the key ideas in our approach.

Thursday October 9, 15:00, room MS.04

Caroline Series (Warwick)

The diagonal slice of Schottky space

Abstract: (Joint work with Ser Peow Tan and Yasushi Yamasita.) We look at representations of the free group on two generators $X$ and $Y$ into $\SL(2,\CC)$ for which $X$, $Y$, and $XY$ have equal trace. We explain how the three-fold symmetry and Keen-Series pleating rays can be used to find those trace values for which the group is free and discrete (a Schottky group) so that the quotient manifold is a genus-two handlebody.

We also find the "Bowditch set", consisting of those representations for which there are only finitely many primitive elements whose trace has absolute value at most $2$ and none is parabolic or elliptic. In contrast to the case of quasifuchsian punctured torus groups originally studied by Bowditch, computer graphics show that this set is significantly different from the discreteness locus.

Thursday October 16, 15:00, room MS.04

Andy Putman (Rice)

The Tits building, class numbers, and the top-dimensional cohomology of $\SL(n, \calO_K)$

Abstract: (Joint work with Tom Church and Benson Farb.) Let $\calO_K$ be the ring of integers in an algebraic number field $K$. The cohomology groups of $\SL(n,\calO_K)$ and $\GL(n,\calO_K)$ play important roles in geometry and number theory. A basic theorem of Borel calculates $H^k(\SL(n,\calO_K);\QQ)$ in the "stable range": that is, when $n$ is large compared to $k$. However, little is known outside the stable range. In this talk, we relate the arithmetic properties of $\calO_K$ to the high-dimensional cohomology groups of $\SL(n,\calO_K)$. Roughly stated, our theorems show that if $\calO_K$ is sufficiently complicated then there must be a large amount of unstable cohomology in the top degree, and conversely if $\calO_K$ is simple enough then there must be no cohomology in the top degree. The keys to our proofs are some new results about the "Steinberg module": the cohomology of the boundary in the Borel--Serre bordification of the associated symmetric spaces.

Thursday October 23, 15:00, room MS.04

Andrew Duncan (Newcastle)

Subgroups of the automorphism groups of partially commutative groups (RAAGs)

Abstract: The class of partially commutative ($pc$) groups contains both finitely generated free and free Abelian groups: as well as everything which lies naturally between these two extremes. Their automorphism groups can thus be seen as lying between general linear groups and the groups $\Aut(F_n)$: both of which have a rich subgroup structure, about which a good deal is known. Decompositions of the automorphism group of an arbitrary $pc$ group, along various subgroups, are described in this talk. The focus is on subgroups of the automorphism group arising as stabilisers of certain subgroups of the $pc$ group, determined naturally by the underlying graph.

Thursday October 30, 15:00, room MS.04

Peter Kropholler (Southampton)

Finiteness conditions for graph-wreath products

Abstract: The graph-wreath product of groups combines the idea of wreath product with that of graph product in a natural way. Given a simple graph and a group \(A\) one can construct a group \(B\) using copies of \(A\) at the vertices of the graph, commuting when two vertices are joined. If \(H\) is a group acting on the graph then the semidirect product of \(B\) and \(H\) is a natural group to consider. Homological finiteness conditions for this group can be established from an understanding of those of \(A\) and \(H\) together with an understanding of cliques in the graph. The methods use infinite Davis complexes and applications include some interesting examples of amenable groups based on Houghton's groups. Thompson's groups also provide examples for which the construction produces groups with strong finiteness conditions.

Thursday November 6, 15:00, room MS.05

Tara Brendle (Glasgow)

Combinatorial models for mapping class groups

Abstract: (Joint work with Dan Margalit.) In 1997 Ivanov proved that the automorphism group of the complex of curves associated to a surface \(S\) is isomorphic to the (extended) mapping class group \(\Mod(S)\), for most surfaces. His work generated a flurry of activity, with similar results obtained by several different authors for various other simplicial complexes associated to surfaces. Ivanov then posed a "metaconjecture" stating that every "sufficiently rich" complex associated to a surface \(S\) has \(\Mod(S)\) as its group of automorphisms. In this talk we will discuss a resolution of Ivanov's metaconjecture for a wide class of complexes.

Thursday November 6, 16:00, room MS.05

Mark Hagen (Michigan)

Virtually special free-by-\(\ZZ\) groups

Abstract: (Joint work with Dani Wise.) Here are two questions about proper, cocompact actions on \(\CAT(0)\) spaces, posed by Gromov and Bridson respectively: (1) Which word-hyperbolic groups admit such actions? (2) Which free-by-cyclic groups admit such actions? This talk is on a positive answer to the "intersection" of these two questions. More precisely, we prove that if \(G\) is a word-hyperbolic group that is (finitely-generated free)-by-cyclic, then \(G\) acts freely and cocompactly on a \(\CAT(0)\) cube complex. This result has interesting consequences beyond showing that \(G\) is \(\CAT(0)\). Notably, together with Agol's theorem on virtual specialness of hyperbolic cubical groups, this shows that \(G\) is linear over the integers.

Thursday November 13, 15:00, room MS.04

Wolfgang Lueck (Haussdorff)

Hyperbolic groups with spheres as boundary

Abstract: (Joint work with Bartels and Weinberger.) Let $G$ be a torsion-free hyperbolic group and let $n$ be an integer, at least six. We prove that $G$ is the fundamental group of a closed aspherical manifold $M$, of dimension $n$, if and only if the boundary of $G$ is homeomorphic to an $(n-1)$-dimensional sphere. In this case $M$ is unique up to homeomorphism.

Thursday November 20, 15:00, room MS.04

Stergios Antonakoudis (Cambridge)

The complex geometry of Teichmüller spaces and symmetric domains

Abstract:From a complex analytic perspective, Teichmüller space - the universal cover of the moduli space of Riemann surfaces - is a contractible bounded domain in a complex vector space. Likewise, bounded symmetric domains arise as the universal covers of locally symmetric varieties (of non-compact type). In this talk we will study isometric maps between these two important classes of bounded domains equipped with their intrinsic Kobayashi metric.

Thursday November 27, 15:00, room MS.04

Agelos Georgakopoulos (Warwick)

The planar Cayley graphs are effectively enumerable

Abstract: The discrete groups acting nicely on the plane, called planar discontinous groups, are a classical topic, closely related to e.g. surface groups. Planar discontinous groups have Cayley graphs that can be embedded in the plane without accumulation points of vertices, like for example the familiar tessellations of the hyperbolic plain, or Escher's tilings. There are however planar Cayley graphs all embeddings of which must have accumulation points of vertices. These graphs, and their groups, were far less well understood, and an open question due to Droms et. al. was whether they can be effectively enumerated.

We settle this question in the affirmative. The main part of our proof is to show that such groups admit nice presentations where the planarity `can be seen' in an automatic way.

Thursday December 4, 15:00, room MS.04

Jessica Banks (Hull)

The Birman exact sequence for $3$-manifolds

Abstract: This talk will consider the Birman exact sequence for a $3$-manifold $M$. In particular, we will show there is an exact sequence $1 \to \pi_1(M, p) \to \Mod(M, p) \to \Mod(M) \to 1$ unless $M$ is foliated by circles. We will also discuss what can be said when $M$ is of this form.