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13
votes
0answers
319 views

Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
11
votes
0answers
166 views

Must nonunit in group algebra of free group generate proper two-sided ideal?

Let $F$ be a free group and $k$ be a field. If $x$ is an element of the group algebra $k[F]$ that is not a unit (equivalently, that is not a nonzero scalar multiple of an element of $F$), must the 2-...
6
votes
0answers
157 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 ...
5
votes
0answers
173 views

Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology. Is it possible that $\alpha(H) \...
4
votes
0answers
97 views

Example of “exotic” verbal subgroups of free groups

This will be an ambiguous question. I am interested in various examples that appear in the literature of verbal subgroups of free groups, but which are not part of the "classical examples" like ...
4
votes
0answers
184 views

Primitive elements in a free group

Let $F$ be a free group of rank $\aleph_0$ and let $H$ be a subgroup for which there exists some $a \in F$ such that $\langle H \cup \{a\} \rangle = F$. Must there be some free basis $B$ for $F$ and a ...
4
votes
0answers
127 views

Is a finitely generated subgroup of a free profinite group virtually a retract?

Let $F$ be a nonabelian finitely generated free profinite group, and let $H \leq F$ be a finitely generated closed subgroup. Must there be some open subgroup $H \leq U \leq F$, and a closed normal ...
4
votes
0answers
166 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 ...
4
votes
0answers
212 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
293 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
184 views

pro-p dense subgroup in the free group

Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental ...
3
votes
0answers
92 views

Geometric automorphism of free group respect to nonorientable suface

An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...
3
votes
0answers
169 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
votes
0answers
87 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 $N \cap H = \{1\}$?
3
votes
0answers
262 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
185 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
80 views

Rank gradient in free products amalgamating a finite subgroup

Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$. Must $A *_C B$ have positive rank gradient? See Which 3-manifolds have ...
2
votes
0answers
102 views

Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to \mathrm{GL}_k(\mathbb{Z}/p\mathbb{...
2
votes
0answers
115 views

Equalizer in Free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and for each $1\leq i\leq n$, let a homomorphism $d_i:F_n\to F_{n-1}$ be defined as follows: $d_i(x_r)=x_r$, if $i>r$; $d_i(x_r)=1$, if $...
2
votes
0answers
105 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
114 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
356 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) := \...
1
vote
0answers
56 views

Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
1
vote
0answers
70 views

Salvaging Howson's theorem for free profinite groups

This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem? Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
1
vote
0answers
71 views

Free profinite products

Let $F$ be a nonabelian finitely generated free profinite group, and let $x \in F$. Must there be some $1 \neq y \in F$ such that $\langle x,y \rangle$ is isomorphic to the free profinite product of $\...
1
vote
0answers
148 views

Can a profinite completion be free pro-p?

Is there a prime number $p$ and a finitely generated residually finite group whose profinite completion is a free pro-$p$ group on a nonempty finite set? Thanks to YCor we see that we cannot take the ...
1
vote
0answers
69 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 $p$...
1
vote
0answers
134 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 ...
1
vote
0answers
80 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 ...
1
vote
0answers
192 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 \...
1
vote
0answers
162 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 ...
0
votes
0answers
95 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 $\Phi \cap N = \{1\}$?
0
votes
0answers
72 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 $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
0
votes
0answers
85 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 ...