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115 views

### Is the free abstract group residually of rank d > 2?

Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements.
Is ...

**5**

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**0**answers

134 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**

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137 views

### Is there a nontrivial profinite word which is trivial in any group with at most d generators?

Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...

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179 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 ...

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171 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**

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100 views

### Growth of the number of generators in hyperbolic groups

Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...

**3**

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71 views

### A Karrass-Solitar or Ivanov-Schupp for profinite groups

Let $F$ be a nonabelian free profinite group, $H \leq_c F$ finitely generated with $[F:H] = \infty$. Must there be some $\{1\} \neq N \lhd_c F$ such that $N \cap H = \{1\}$?

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188 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 ...

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169 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**

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85 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?

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100 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 $F_n$)? ...

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283 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) := ...

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45 views

### Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let ...

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107 views

### An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...

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62 views

### Bases for free pro-p groups

Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...

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168 views

### Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...

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132 views

### Accessible subgroups of free groups

Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...

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54 views

### Bases of surface groups with length restrictions

This question asks for a generalization of Bases of surface groups following the notation and definitions given therein.
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, $B$ a surface basis of ...

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74 views

### Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?

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66 views

### A bound on the size of the center

Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

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80 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 ...