Geometry and Topology Seminar

Thursday January 23, 15:00, room MS.03

Brian Bowditch (Warwick)

Products of hyperbolic spaces

Abstract: We describe a variation of a result of Kapaovich, Kleiner and Leeb. This says that a quasi-isometric embedding of a finite product of hyperbolic spaces in another respects the decomposition into factors, up to bounded distance and permutation of factors, provided we assume that the factors in the domain are "bushy". There is also a generalisation allowing for Euclidean factors.

Thursday January 30, 15:00, room MS.03

Karen Vogtmann (Warwick)

The geometry of Outer space

Abstract: Outer space was introduced in the early 1980's as a tool for studying the group \(\Out(F_n)\) of outer automorphisms of a finitely-generated free group. It is a contractible space on which \(\Out(F_n)\) acts, with nice stabilizers, and can be thought of as analogous to a symmetric space (with the action of a non-uniform lattice) or the Teichmüller space of a surface (with the action of the mapping class group of the surface). Much progress has been made on understanding the topology of Outer space and of its quotient by \(\Out(F_n)\); but its geometry was largely unexplored until recently. Now through the efforts of many people a metric theory is emerging, resulting in new information about \(\Out(F_n)\) as well as elegant new proofs of older results, and strengthening the analogy between the classical theories of symmetric spaces and Teichmüller spaces. I will describe the basics of this theory, then focus on some striking recent work of Bestvina-Feighn and Handel-Mosher, who use this new geometry to prove that certain simplicial complexes naturally associated to free groups have negative curvature in the sense of Gromov.

Thursday February 6, 15:00, room MS.03

Alessandro Sisto (ETH)

Bounded cohomology via hyperbolically embedded subgroups

Abstract:(Joint with Roberto Frigerio and Beatrice Pozzetti.) Hyperbolically embedded subgroups have been defined by Dahmani-Guirardel-Osin and they provide a common perspective on (relatively) hyperbolic groups, mapping class groups, \(\Out(F_n)\), CAT(0) groups and many others. Quasi-cocycles on hyperbolically embedded subgroups can be extended to the ambient group, and as a consequence one gets that the second and third bounded cohomology of, say, mapping class groups are infinite dimensional. I will also discuss the fact that certain such extended quasi-cocycles (of dimension 2 and higher) "contain" the information that \(H\) is hyperbolically embedded in \(G\).

Thursday February 13, 15:00, room MS.03

Chris Wendl (UCL)

When is a Stein manifold merely symplectic?

Abstract: Stein manifolds are objects originating in complex geometry that also naturally carry symplectic structures. In recent years, the study of Stein structures has increasingly been dominated by the question of "rigid vs. flexible": on the flexible side, the so-called "subcritical" Stein manifolds satisfy an h-principle, so their Stein homotopy type is determined by the homotopy class of the underlying almost complex structure, and all "interesting" invariants of such structures vanish. At the other end of the spectrum, one should expect to find pairs of Stein manifolds that are symplectomorphic but not Stein deformation equivalent, though no examples are yet known. In this talk, I will explain where NOT to look for examples: in complex dimension 2, there is a large class of Stein domains that exist somewhere between rigid and flexible, meaning that while the h-principle does not hold in any strict sense, their Stein deformation type is completely determined by their symplectic deformation type. This result depends on some joint work with Sam Lisi and Jeremy Van Horn-Morris involving the relationship between Stein structures and Lefschetz fibrations, which can sometimes be realised as foliations by J-holomorphic curves.

Thursday February 20, 15:00, room MS.03

Jim Anderson (Southampton)

Subgroups of classical and non-classical Schottky groups

Abstract: Schottky groups are Kleinian groups, in this case acting on \(\HH^3\), generated by a collection of loxodromic Moebius transformations pairing Jordan curves that bound a common domain in the Riemann sphere \(\overline{\CC}\). Marden (1978) gave a non-constructive proof of the existence of Schottky groups that do not arise by pairing circles in \(\overline{\CC}\). Yamamoto (1991) gave an explicit construction in rank 2. In this talk, we consider the extent to which subgroups of classical (i.e., constructed by pairing circles) and non-classical Schottky groups are themselves classical and non-classical.

Thursday February 27, 15:00, room MS.03

Moritz Rodenhausen (Warwick)

Centralisers of polynomially growing automorphisms of free groups

Abstract: I first recall definitions of graphs of groups and Dehn twist automorphisms of free groups as well as efficient Dehn twists in the sense of Cohen and Lustig. I state a structure theorem about centralisers of Dehn twists in \(\Out(F_n)\) and \(\Aut(F_n)\), which is joint work with Ric Wade. This shows finiteness property VF for these centralisers. Finally, I sketch an extension of this result to centralisers of "higher" Dehn twists, which include all polynomially growing automorphisms up to passing to powers.

Thursday March 6, 15:00, room MS.03

Dmitri Panov (Kings)

Definite connections

Abstract: (Joint work with Joel Fine.) A definite connection is an $\SO(3)$ connection over a $4$-manifold, whose curvature is non-zero on every tangent $2$-plane. Given such a connection, the associated $2$-sphere bundle is naturally a symplectic manifold. In this talk I will be mainly interested in definite connections invariant under a circle action, in which case the corresponding symplectic six-manifold is "Fano". I will explain why only $S^4$ and $\CP^2$ admit an $S^1$-invariant definite connection.

Thursday March 13, 15:00, room MS.03

Moon Duchin (Tufts)

Geometry versus group theory in the Heisenberg group

Abstract: Work of Pansu gives us a geometric approach to studying the large-scale geometry of nilpotent groups, and some recent refinements by Breuillard-Le Donne and Duchin-Mooney give hands-on tools suitable for a very precise study of the Heisenberg group. I'll explain some applications, for instance to the study of the growth function and growth series, and I'll discuss current work with Mike Shapiro on rationality of growth.

Thursday March 13, 16:00, room MS.03

Michael Shapiro (Tufts)

Equations in nilpotent groups

Abstract: An equation in a group is a word $w$, where some of the letters of $w$ are constants (elements of the group) and some of the letters of $w$ are variables. We can solve the equation if we can assign group elements to the variables to arrange $w = 1$. A system of equations is a collection of such words. Given a group, we would like an algorithm that determines whether any equation or any system of equations has a solution.

We show that there is an algorithm which does this for any single equation in the Heisenberg group. We also show that for any non-Abelian free nilpotent group there is no algorithm which decides systems of equations. Further there is no algorithm which decides a single equation in any free nilpotent group of step 3 or more.