
Geometry and Topology Seminar

Warwick Mathematics Institute, Term II, 2015-2016

 Thursday January 14, 15:00, room MS.03 Caroline Series (Warwick) Continuous motions of limit sets Abstract: A Kleinian group is a discrete group of isometries of hyperbolic $3$-space. Its limit set, contained in the Riemann sphere, is the set of accumulation points of any orbit. In particular the limit set of a closed, hyperbolic, surface group $F$ is the unit circle. If $G$ is a Kleinian group abstractly isomorphic to $F$, then there is an induced map, known as a Cannon-Thurston (CT) map, between their limit sets. More precisely, the CT-map is a continuous equivariant map from the unit circle to the Riemann sphere. Suppose now $F$ is fixed while $G$ varies. We discuss work with Mahan Mj about the behaviour of the corresponding CT-maps, viewed as maps from the circle to the sphere. We explain how a simple criterion for the existence of a CT-map can be adapted to establish conditions on convergence of a sequence of groups $G_n$ under which the corresponding sequence of CT-maps converges uniformly to the expected limit. We also discuss an example which shows that under certain circumstances, CT-maps may not even converge pointwise.

 Thursday January 28, 15:00, room MS.04 Beatrice Pozzetti (Warwick) Ultralimits of maximal representations Abstract: (Joint work with Marc Burger.) A representation of the fundamental group of an hyperbolic surface in the symplectic group $\Sp(2n,\RR)$ is called maximal if it maximize the so-called Toledo invariant. Maximal representations form interesting and well studied components of the character variety generalizing the Teichmuller component, that is encompassed in the case $n = 1$. Given an unbounded sequence of maximal representations one naturally gets an action on an affine building. I will describe geometric properties of such actions, dealing in particular with the structure of elements acting with a fixed point.

 Thursday February 4, 15:00, room MS.04 Patricia Cahn (Max Planck) Knots transverse to a vector field Abstract: (Joint work with Vladimir Chernov.) We study knots transverse to a fixed vector field $V$ on a 3-manifold $M$, up to the corresponding isotopy relation. Such knots are equipped with a natural framing. Motivated by questions in contact topology, it is natural to ask whether two $V$-transverse knots which are isotopic as framed knots and homotopic through $V$-transverse immersed curves must be isotopic through $V$-transverse knots. When $M$ is $\RR^3$ and $V$ is the vertical vector field the answer is yes. However, we construct examples which show the answer to this question can be no in other 3-manifolds, specifically $S^1$-fibrations over surfaces of genus at least 2. We also give a general classification of knots transverse to a vector field in an arbitrary closed oriented 3-manifold $M$. We show this classification is particularly simple when $V$ is the co-orienting vector field of a tight contact structure, or when $M$ is irreducible and atoroidal. Lastly, we apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges.

 Thursday February 11, 15:00, room MS.04 Andrew Brooke-Taylor (Bristol) The complexity of a complete knot invariant Abstract: (Joint work with Sheila Miller.) Quandles are algebraic structures which Joyce showed could be used as complete invariants for tame knots. However, there is some dissatisfaction with them as invariants, as it heuristically seems difficult to determine whether two quandles are isomorphic. We have shown that for arbitrary countable quandles, this isomorphism problem is indeed as complex as possible, in the sense of Borel reducibility. In contrast, tame knot isomorphism is trivial in this setting, leaving the door open to finding more tractable complete invariants for knots.

 Thursday February 18, 15:00, room MS.04 Oscar Randal-Williams (Cambridge) Tautological rings for high-dimensional manifolds Abstract: (Joint work with Ilya Grigoriev and Soren Galatius.) The cohomology of the classifying space $\BDiff(M)$ of the group of diffeomorphisms of a manifold $M$ may be considered as the ring of characteristic classes of smooth fibre bundles with fibre $M$. This ring is difficult to understand, but when $M$ is an orientable surface the close connection between $\BDiff(M)$ and the moduli space of Riemann surfaces means that a lot is known. In this case, algebraic geometers have found it productive to focus not on all the cohomology but a certain subring, the "tautological ring", containing the geometrically interesting classes. One can make a similar definition for manifolds of higher dimension. I will explain all these terms, and discuss some recent results on the large scale structure of these tautological rings.

 Thursday February 25, 15:00, room MS.04 Arnaud Chéritat (Toulouse/Bordeaux/CNRS) Self-similarity of Siegel disks Abstract: A famous result of McMullen states that the boundaries of Siegel disks (for certain polynomials) are self-similar at the critical point. The proof involves quasiconformal surgery on a Blacshke model where the Siegel disk is a euclidean circle. Elementary numerical experiments suggest a form of self-similarity also holds for some Siegel disks of exponentials. In this case the Siegel disk boundary is not locally connected, hence it is not clear that McMullen's approach can adapt. A more promising approach is via the use of cylinder renormalization. The latter is well developed in the case of near parabolic quadratic polynomials. This talk will report on the work in progress for, on one hand reproving self-similarity with cylinder renormalization, on the other hand extending cylinder renormalization to $z^d + c$ and hopefully the exponential.

 Thursday March 3, 15:00, room MS.04 Daniel Woodhouse (McGill) A geometric proof of the non-separability of a 3-manifold group Abstract: A group $G$ is residually finite if every non-trivial element survives in a finite quotient. A subgroup $H$ is separable if every element of $G$ that is not in $H$ survives in a finite quotient that eliminates $H$. I will discuss these notions, focusing on the topological criterion for separability of Scott and how it relates to the Virtual Haken Conjecture for 3-manifolds. A proof that the 3-manifold group of Burns-Karass-Solitar is non-separable will be presented using pictures. If there is time I will indicate how this proof relates to my work cubulating tubular groups.

 Thursday March 3, 16:00, room MS.04 Francis Brown (Oxford) Special automorphisms of free Lie algebras Abstract: This talk will be a gentle introduction to some ideas from Grothendieck-Teichmüller theory. I will explain how, using some analysis and some ideas from the theory of periods, one can construct derivations on free Lie algebras with remarkable combinatorial properties. No prior familiarity with Grothendieck-Teichmüller theory, or the theory of periods, will be needed.

 Thursday March 10, 15:00, room MS.04 Matthew Tointon (Paris-Sud) Nilpotent approximate groups Abstract: There has been much recent study of the so-called 'approximate subgroups' of a group. These are, roughly speaking, sets obtained when one relaxes the definition of a subgroup to include sets that are only 'approximately closed' under the group operation. Approximate groups arise naturally in many geometric settings, and their study has striking and far-reaching consequences in a number of fields, including differential geometry and geometric group theory. I will give a brief overview of the field with a particular focus on geometric applications (including some obtained recently in joint work with Emmanuel Breuillard). I will then present a structure theorem for the approximate subgroups of a nilpotent group.

 Thursday March 10, 16:00, room MS.04 Javier Aramayona (Madrid) On the automorphism group of a right-angled Artin group Abstract: (Joint work with Conchita Martinez-Perez.) Automorphism groups of right-angled Artin groups form an interesting class of groups, as they interpolate" between the two extremal cases of $\Aut(F_n)$ and $\GL(n,\ZZ)$. In this talk we will discuss some conditions on a simplicial graph which imply that the automorphism group of the associated right-angled Artin group has (in)finite abelianization. As a direct consequence, we obtain families of such automorphism groups that do not have Kazhdan's property (T).

 Thursday March 17, 15:00, room MS.04 Carmen Rovi (Max Planck) The signature modulo eight of a fibration Abstract: In this talk we shall be concerned with the residues modulo four and modulo eight of the signature $\sigma(M)$ in $\ZZ$ of an oriented $4k$-dimensional geometric Poincare complex $M^{4k}$. The precise relation between the signature modulo eight, the Arf invariant and the Brown-Kervaire invariant will be given. Furthermore we shall discuss how the relation between these invariants can be applied to the study of the signature modulo eight of a fibration. In particular it had been proved by Meyer in 1973 that a surface bundle has signature divisible by four. This was generalized to higher dimensions by Hambleton, Korzeniewski and Ranicki in 2007. I will explain two results from my thesis concerning the signature modulo eight of a fibration: firstly under what conditions can we guarantee divisibility of the signature by eight and secondly what invariant detects non-divisibility by eight in general.

Information on past talks. This line was last edited Thursday, 30 July 2015 19:23:35 BST.