\( \newcommand{\SL}{\operatorname{SL}} \newcommand{\Pin}{\operatorname{Pin}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Out}{\operatorname{Out}} \newcommand{\CC}{\mathbb{C}} \newcommand{\HH}{\mathbb{H}} \newcommand{\ZZ}{\mathbb{Z}} \)

Groups and Geometry in the South East


20 - 21 February, 2014

Organized by Martin Bridson and Henry Wilton with assistance from Saul Schleimer.

If you are interested in attending, please register. Limited funds are available to UK academics for housing and travel.

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 20, 16:00, room MS.03

Jongil Park (SNU)

How to construct \(4\)-manifolds with \(b_2^+=1\)

Abstract: One of the fundamental problems in the topology of \(4\)-manifolds is to classify all smooth and symplectic manifolds, and complex surfaces, homeomorphic to a given topological \(4\)-manifold. Even though gauge theory has been very successful in this direction, the complete answer in the case of manifolds with \(b_2^+ = 1\) is still far out of reach.

The aim of this talk is to briefly review known techniques to produce simply connected \(4\)-manifolds with \(b_2^+ = 1\) (repsectively, \(p_g=0\)) in the smooth and symplectic categories (respectively, in the complex category).

Friday February 21, 10:00, room MS.04

Seonhee Lim (SNU)

Colorings of trees and their subword complexity

Abstract:(Joint with D. Kim and S. Lee.) We will introduce the subword complexity of colorings of trees, generalizing the subword complexity in symbolic dynamics. Sturmian colorings, which are colorings of minimal unbounded subword complexity, are always induced from a coloring of a quotient graph which is a ray. We will then discuss more general colorings.

Friday February 21, 11:00, room MS.04

Sanghyun Kim (KAIST and SNU)

RAAGs in braids

Abstract: (Joint with Thomas Koberda.) We show that every right-angled Artin group (RAAG) embeds into a RAAG defined by the opposite graph of a tree. We then embeds, by quasi-isometries, an arbitrary RAAG into a pure braid group and also into the area-preserving diffeomorphism groups of the disk and of the sphere; this answers questions due to Crisp-Wiest and to M. Kapovich.

Friday February 21, 13:15, room MS.04

Enric Ventura (UPC)

On the difficulty of inverting automorphisms of free groups.

Abstract: (Joint work with P. Silva and M. Ladra.) We introduce a complexity function \(\alpha\) (respectively \(\beta\)) to measure the maximal possible gap between the norm of an automorphism (respectively an outer automorphism) of a finitely generated group \(G\), and the norm of its inverse. We shall concentrate in the case of free groups \(F_r\) and prove some results about the growth of these functions \(\alpha_r\) and \(\beta_r\): the gap \(\alpha_2\) is quadratic and \(\beta_2\) is linear. For higher rank, we will give polynomial lower bounds for both functions, and a polynomial upper bound for \(\beta_r\). The lower bounds simply require manipulating automorphisms and some counting techniques. The proof of the upper bound makes use of a recent result by Algom-Kfir and Bestvina about the asymmetry of the metric in the Outer Space.

Friday February 21, 14:30, room MS.04

Pavel Zalesskii (Brasilia)

Pro-p ends

Abstract: We shall discuss a pro-p analogue of Stallings' theory of ends.

Friday February 21, 16:00, room B3.03 (Colloquium)

Tim Riley (Cornell)

Hyperbolic groups, Cannon-Thurston maps, and hydra

Abstract: Groups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re-imagining of Hercules' battle with the hydra, where wildness is found in properties of "Cannon-Thurston maps" between boundaries. Also, I will give examples where this map between boundaries fails to be defined.

Partly supported by LMS Scheme Three, by the Warwick-SNU International Partnership Fund, and by an EPSRC Warwick Platform Grant.