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5
votes
0answers
126 views

Dynamics of virtual automorphisms of free group

The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism. Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the ...
4
votes
0answers
174 views

Schreier's formula and descending chains

For a group $G$ we denote by $d(G)$ the cardinality of a smallest set of generators. A finitely generated group $G$ is said to satisfy Schreier's formula if for every subgroup $H \subseteq G$ of ...
4
votes
0answers
154 views

Conjugacy classes of elements in free groups. One-variable equations.

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 ...
3
votes
0answers
175 views

Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...
3
votes
0answers
164 views

Generating set for abelianization of “mod $p$ commutator subgroup” of a free group

Let $F_n$ be a free group on $n$ letters, and fix some prime $p \geq 2$. Define $$K_{n,p}=\text{ker}(F_n \rightarrow H_1(F_n;\mathbb{Z}/p))$$ and $$V_{n,p} = H_1(K_{n,p};\mathbb{Q}).$$ For $x \in ...
2
votes
0answers
78 views

Free profinite completions

Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?
2
votes
0answers
98 views

Elements of minimal length in normal closures of elements in free groups

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 $F_n$)? ...
2
votes
0answers
258 views

Stable commutator length of elements in free groups.

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) := ...
0
votes
0answers
78 views

Bases and transversals

Let $F$ be a free finitely generated group, $L \leq H \leq F$ subgroups of finite index. Given bases $B$ of $F$ and $C$ of $H$, must there be a Schreier transversal $T$ for $H$ in $F$ such that the ...