
Using an analogue of MakaninRazborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (rightangled 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 kgenerator 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 kgenerator 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 kgenerator 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 quasiisometric rigidity for lattices in SOL, EskinFisherWhyte develop a technique of "coarse differentiation" to prove a theorem on the structure of quasiisometries of SOL. This structure theorem combined with a theorem on bilipshitz maps of the real line completes the proof of quasiisometric rigidity for lattices in SOL. EskinFisherWhyte and Peng use similar techniques to extend this structure theorem to quasiisometries of more general solvable Lie groups. We will present the ingredients needed to complete the proof of quasiisometric 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 languagetheoretic conditions, in particular lying in the classes of starfree, 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 rightangled Coxeter groups are elementarily equivalent if and only if they are isomorphic.
We discuss the dynamics of free action of a nonorientable surface group on an Rtree. 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 "Wvalued 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 pseudoAnosov. 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 NPcomplete.
This is joint work with O. Kharlampovich, I. G. Lysenok and A. G. Myasnikov.
We study the computational complexity of the Word Problem (WP) in free solvable groups S_{r,d}, where r ³ 2 is the rank and d ³ 2 is the solvability class of the group. It is known that the Magnus embedding of S_{r,d} into matrices provides a polynomial time decision algorithm for WP in a fixed group S_{r,d}. Unfortunately, the degree of the polynomial grows together with d, so the uniform algorithm is not polynomial in d. In this paper we show that WP has time complexity O(rn log_{2} n) in S_{r,2}, and O(n^{3} rd) in S_{r,d} for d ³ 3. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in S_{r,2} is NPcomplete.