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This is the first in a planned series of papers giving an alternate approach to Zlil Selas work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G,F) where G is a finitely generated group and F is a non-abelian free group.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of n-tracks of length n, which is a structure like Z+Z, and prove its existence in the non-h
The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin
We prove that for geometrically finite groups cohomological dimension of the direct product of a group with itself equals 2 times the cohomological dimension dimension of the group.
Let $Gamma$ be the fundamental group of a surface of finite type and Comm$(Gamma)$ be its abstract commensurator. Then Comm$(Gamma)$ contains the solvable Baumslag--Solitar groups $langle a ,b : a b a^{-1} = b^n rangle$ for any $n > 1$. Moreover, the
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.