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Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...

My broad question is regarding the lengths of (reduced) words in a subgroup of a free group. As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...

Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$. What is the ...

I will pose the question in the form in which it originally appeared to me: Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\...

For $i+j+k=N$, consider the permutation $\Pi_{i,j,k}\in S_N$, which keeps the numbers $0,\ldots,i-1$ fixed, and exchanges the numbers $i,\ldots,i+j-1$ with the numbers $i+j,\ldots,i+j+k-1$. $$\Pi_{i,j,...