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_ …

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 …

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{s …

As is well known, the conjugacy classes of the free group $F_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two …

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 fini …

Let $F_n$ be a free group, $u, v \in F_n$, where $[u,v] \neq 1 $. Trying to show that the following equation does not have a solution in $F_n$ : $$ [v,x] = [u,v] .$$ Any ideas a …

Let $F_m$ be the free group with $m$ generators $S:=\{x_1,\dots, x_m\}$. I am interested in the following quantity $$ F(n):=\frac{|\{w\in [F_m:F_m]: \|w\|_S\leq n\}|}{|B_S(n)|} $$ …

We write $cl$ for the commutator length, i.e. the least number of commutators which multiply to a given element of a group. Given an element $g$ in the commutator subgroup of the …

The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operatio …

Consider the group $G=\langle x_1,x_2,x_3|x_1^2,x_2^2,x_3^2\rangle$. Using a slightly modified version of S. Ivanov's proof here that free groups are residually finite, I can show …

Let $F_n$ be the free group generated by $x_i$, for $1\leq i\leq n$. Let $a_i$ be some elements of $F_n$, also for $1\leq i\leq n$. Is there a nice way to tell when the list ${a_i^ …

For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ …

I am not sure yet about what I exactly need to prove, but I guess I can formulate a rough statement similar to the following: *Suppose $w\in F_2$ is a primitive word whose length …

Let me recall the definition which seems the most standard of Fricke polynomials. Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exis …

I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me. Let $F$ be a free group with $n$ generators $x_ …