# Lecture Series September

## Free groups, automorphisms, hyperbolic groups

Lecture Series III: Random Groups and Surfaces

Speaker: Gaven Martin, Massey University, New Zealand

Dates:

(1) Monday, September 24, 2 p.m.

(2) Wednesday, September 26, 2 p.m.

(3) Friday, September 28, 2 p.m.

Abstract:

This series of talks will relate a project I have been thinking about for the last year or so. For the classical finite groups there are results of Kantor / Lubotzky (and Liebeck / Shalev otherwise): Put the uniform distribution on PSL(2,Fq). Then as q → ∞, Pr {<f,g> = PSL(2,Fq), f, g ∈ PSL(2,Fq)} → 1. These results have all sorts of applications. We consider what happens for infinite groups? In particular, the geometrically interesting groups PSL(2,R) or on PSL(2,C) of isometries of hyperbolic two and three dimensional space. We discuss a geometrically natural probability measure on these groups and then by selecting finitely many generators g1,g2,...,gn at random with respect to this measure we define a "random" group of isometries. The probability measure we identify in effect establishes an isomorphism between random n-generators groups and collections of n random pairs of arcs on the circle offering natural generalisations to higher dimensions. A random (two generator) group <f,g> is not discrete iff <f,g> = PSL(2,R) in which case we say f, g generate). We aim to estimate the likely-hood that a random group is discrete, calculate the expectation of their associated parameters, the expected topology (genus) and geometry. Then conditioned by discreteness we investigate random (Riemann) surfaces. These will all have a free group of rank at least two as a fundamental group. As an example a random two generator group of finite co-area yields a triply punctured sphere (with probability 2/3) or a punctured torus (with probability 1/3). Both have fundamental group the free group of rank 2. But in the latter case one is able to investigate more subtle invariants such as the base eigenvalue of Laplacian or dimension of limit sets, shortest geodesics and distributions in Teichmüller space. There is nothing really deep or challenging going on here, you’ll get to see some of the basic aspects of the geometry of discrete groups and maybe learn a little about hyperbolic geometry, Riemann surfaces and probability. You’ll only need to know the Riemann mapping theorem!

**Lecture Notes: Lecture I **

**Lecture Notes: Lecture II **

**Lecture Notes: Lecture III **

Lecture Series II: Surface subgroups of graphs of free groups

Speaker: Henry Wilton, Cambridge

Dates:

(1) Monday, September 24, 10 a.m.

(2) Wednesday, September 26, 10 a.m.

(3) Friday, September 28, 10 a.m.

Abstract:

Hyperbolic groups, introduced by Gromov and others in the 1980s, are discrete groups that satisfy a coarse negative curvature condition. The simplest examples are free groups and surface groups (i.e. the fundamental groups of negatively curved surfaces). Around the time of their introduction, Gromov famously asked a natural question: must a hyperbolic group G contain a surface subgroup, unless it has a free subgroup of finite index? These lectures will describe recent progress in the case when G is the fundamental group of a graph of free groups (for instance, an amalgamated product F1 *H F2). In particular, I’ll describe some of the tools that go into the proof: Whitehead’s algorithm for recognising basis elements of free groups; JSJ decompositions (inspired by 3-manifold topology); and linear programming (as in Calegari’s work on stable commutator length in free groups).

**Lecture notes: Lecture I**

**Lecture notes: Lecture II**

**Lecture notes: Lecture III**

Lecture Series I: Automorphism groups of free groups and their cousins: subgroups, rigidity and complexity

Speaker: Martin Bridson, Oxford

Dates:

(1) Monday, September 17, 3 p.m.

(2) Tuesday, September 18, 10 a.m.

(3) Tuesday, September 18, 2 p.m.

Abstract:

I shall begin with an elementary account of the 3-way analogy between automorphism groups of free groups Aut(F), mapping class groups of surfaces, and SL(n,Z), comparing and contrasting the basic features of these groups and the spaces on which they naturally act. I shall then focus on the rich subgroup structure of these groups, particularly Aut(F), the investigation of which draws one naturally into the study of rigidity phenomena, large-scale geometry, and diverse manifestations of non-positive curvature. Decision problems for Aut(F) and its subgroups will be discussed. The lectures will touch on many different topics and point to many connections, but the key objects will be defined in each case and the lectures will be accessible to a wide audience.