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This is not easy. The complete modern description of the solution sets (Lyndon original result) was refined, for example, in the paper by Remeslennikov and Chiswell, and then a bit later by Myasnikov …

Hi Mark, just wanted to share the solution to this problem as well as a method used. I found quite a powerful tool buried at the end of this paper "A Classification of Fully Residually Free Groups …

Let $F$ be a free group of finite rank, and $p, b \in F$, where $b$ is a root element (i.e. not a proper power). I have a case where $p^{n_k} = V_{n_k}^{-1}b^{-1} V_{n_k} \cdot U_{n_k}^{-1}b U_{n_k}$ …

First of all, wasn't sure what could be a good title for this question. If mods think of a better name, pls feel free to change it... Let $F$ be a (non-abelian) free group of finite rank, a vector $\ …

http://arxiv.org/pdf/math/0611889v4.pdf (page 13) In the above paper by Danny Calegari he says that the result $\text{scl}(g) \geq 1/2$ (i.e. a stable commutator length $\text{scl}(g) := \displayst …

Let $F_n$ be a free group of rank $n$. Let $w\in F_n$ be cyclically reduced. What can be said about the element(s) of minimal length from the $\textit{ncl}(w)$ (normal closure of $w$ in $F_n$)? Unde …

Given a torsion-free hyperbolic group $G$, does there exist a number $n(G)$ such that for any $x,y,z\in G$, $x^n y^n z^n =1$ implies that $x$, $y$, and $z$ commute pairwise? Some musings/questions... …