# Past Algebra Seminar

I will discuss similarities and differences between the geometry of

relatively hyperbolic groups and that of mapping class groups.

I will then discuss results about random walks on such groups that can

be proven using their common geometric features, namely the facts that

generic elements of (non-trivial) relatively hyperbolic groups are

hyperbolic, generic elements in mapping class groups are pseudo-Anosovs

and random paths of length $n$ stay $O(\log(n))$-close to geodesics in

(non-trivial) relatively hyperbolic groups and

$O(\sqrt{n}\log(n))$-close to geodesics in mapping class groups.

We discuss the problem to what extend a group action determines geometry of the space.

More precisely, we show that for a large class of actions measurable isomorphisms must preserve

the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

I will introduce the notion of Kurosh rank for subgroups of

free products. This rank satisfies the Howson property, i.e. the

intersection of two subgroups of finite Kurosh rank has finite Kurosh rank.

I will present a version of the Strengthened Hanna Neumann inequality in

the case of free products of right-orderable groups. Joint work with A.

Martino and I. Schwabrow.

One of the applications of the study of assymptotics of

homology groups in residually free groups of type FP_m is the calculation

of their analytic betti numbers in dimension up to m.

The homological dimension of a group can be computed over any coefficient ring $K$.

It has long been known that if a soluble group has finite homological dimension over $K$

then it has finite Hirsch length and the Hirsch length is an upper bound for the homological

dimension. We conjecture that equality holds: i.e. the homological dimension over $K$ is

equal to the Hirsch length whenever the former is finite. At first glance this conjecture looks

innocent enough. The conjecture is known when $K$ is taken to be the integers or the field

of rational numbers. But there is a gap in the literature regarding finite field coefficients.

We'll take a look at some of the history of this problem and then show how some new near complement

and near supplement theorems for minimax groups can be used to establish the conjecture

in special cases. I will conclude by speculating what may be required to solve the conjecture outright.

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.

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.