I'll discuss some results about lattices in totally

disconnected locally compact groups, elaborating on the question:

which classical results for lattices in Lie groups can be extended to

general locally compact groups. For example, in contrast to Borel's

theorem that every simple Lie group admits (many) uniform and

non-uniform lattices, there are totally disconnected simple groups

with no lattices. Another example concerns with the theorem of Mostow

that lattices in connected solvable Lie groups are always uniform.

This theorem cannot be extended for general locally compact groups,

but variants of it hold if one implants sufficient assumptions. At

least 90% of what I intend to say is taken from a paper and an

unpublished preprint written jointly with P.E. Caprace, U. Bader and

S. Mozes. If time allows, I will also discuss some basic properties

and questions regarding Invariant Random Subgroups.

# Past Algebra Seminar

Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of Aut(G) on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice Γ in G is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of G is connected. I will introduce all necessary notions and try to motivate and explain this result.

Most results about the Cayley graph of a hyperbolic surface group can be replicated in the context of more general hyperbolic groups. In this talk I will discuss two results about such Cayley graphs which I do not know how to replicate in the more general context.

In our 2004 paper, Fritz Grunewald and I constructed the first

pairs of finitely presented, residually finite groups $u: P\to G$

such that $P$ is not isomorphic to $G$ but the map that $u$ induces on

profinite completions is an isomorphism. We were unable to determine if

there might exist finitely presented, residually finite groups $G$ that

with infinitely many non-isomorphic finitely presented subgroups $u_n:

P_n\to G$ such that $u_n$ induces a profinite isomorphism. I shall

discuss how two recent advances in geometric group theory can be used in

combination with classical work on Nielsen equivalence to settle this

question.

We present a new, more conceptual proof of our result that, when a finite group acts on a polynomial ring, the regularity of the ring of invariants is at most zero, and hence one can write down bounds on the degrees of the generators and relations. This new proof considers the action of the group on the Cech complex and looks at when it splits over the group algebra. It also applies to a more general class of rings than just polynomial ones.

I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|.

I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.

There is a well-acknowledged analogy between mapping class

groups and lattices in higher rank groups. I will discuss to which

extent does Margulis's superrigidity hold for mapping class groups:

examples, very partial results and questions.