Using an analogue of Makanin-Razborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (right-angled Artin group) G. Equivalently, we give a parametrisation of Hom(H,G), where H is a finitely generated group.

There are a number of possible notions of constructing k-generator subgroups "at random" from a fixed group G. Given such a process, we can try to understand properties that a random subgroup has. For random subgroups of Thompson's group F, a number of interesting phenomena occur which are not present in other known examples. For example, there are positive densities of many isomorphism classes of k-generator subgroups, rather than there just being one isomorphism class of density 1. I will also describe a persistence phenomenon seen in Thompson's group, where some isomorphism classes of subgroups are present with positive density in the space of k-generator subgroups for all k larger than some K, with respect to one of the natural processes for constructing subgroups at random.

This is joint work with Murray Elder, Andrew Rechnitzer and Jennifer Taback.

In their proof of quasi-isometric rigidity for lattices in SOL, Eskin-Fisher-Whyte develop a technique of "coarse differentiation" to prove a theorem on the structure of quasi-isometries of SOL. This structure theorem combined with a theorem on bilipshitz maps of the real line completes the proof of quasi-isometric rigidity for lattices in SOL. Eskin-Fisher-Whyte and Peng use similar techniques to extend this structure theorem to quasi-isometries of more general solvable Lie groups. We will present the ingredients needed to complete the proof of quasi-isometric rigidity for these more general solvable Lie groups. This involves, among other things, a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces.

The word problem for finitely presented groups was posed in 1911 by Max Dehn. Recent results in complexity theory and cryptography have drawn attention to randomized versions of the word problem as well as to corresponding versions of the conjugacy and isomorphism problems. We survey these developments and discuss current progress.

Let Geo(G,X) be the set of all words over a generating set X of a group G that label geodesic paths in the corresponding Cayley graph for G. For many classes of groups, including word hyperbolic groups, Coxeter groups, and Garside groups, it is known that there are finite generating sets X for which Geo(G,X) is regular. In this talk I will discuss geometric and combinatorial properties of finitely generated groups for which Geo(G,X) satisfies stronger language-theoretic conditions, in particular lying in the classes of star-free, locally excluding, and locally testable languages.

We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $"-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.

We discuss the dynamics of free action of a non-orientable surface group on an R-tree. This is related to linear involutions with flips.

Suppose that all hyperbolic groups are residually finite. Then in relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable.

Joint work with J. Manning (SUNY Buffalo).

A building F is a combinatorial object that come with an "W-valued distance function", where W is a Coxeter group. I will `remind' everyone of the definitions and of two standard ways to give buildings a geometric structure. One is due to Tits, which works very well for spherical and Euclidean buildings, and another is due to Davis, that has been important in geometric group theory. A third approach is something of a hybrid, and it has been used recently in the computation of the compactly supported cohomology of buildings.

This is joint work with Davis, Dymara, Januszkiewicz and Okun.

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.

This is joint work with Matt Clay.

In this talk I shall give a survey of results on elementary theory of f.g. nilpotent groups. Then my work on characterizing groups elementary equivalent to a free nilpotent group of finite rank will be discussed. I shall end the talk with a discussion of certain possible connections between elementary theory and coarse geometry of f.g. nilpotent groups.

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.

This is joint work with O. Kharlampovich, I. G. Lysenok and A. G. Myasnikov.