Questions tagged [free-groups]

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4 votes
1 answer
286 views

Extending primitive systems in free groups

It is a version of a question I asked on Math.StackExchange (no answers yet). Suppose $a_1,a_2,\ldots,a_n,b$ is a primitive system in a non-abelian free group $F$ (a part of a basis of $F$). Let $c \...
9 votes
1 answer
500 views

Shortest almost trivial element of free group [duplicate]

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 ...
12 votes
3 answers
942 views

Road map to learn about $\operatorname{Out}{F_n}$

I'm a last year undergraduate student and I have taken a graduate course in geometric group theory. I'd like to start reading some more advanced stuff in geometric group theory and in particular about ...
2 votes
1 answer
146 views

Almost free group without the Specker group as a subgroup

An Abelian group is almost free whenever every countable subgroup is free Abelian. Famously, the Specker group $\mathbf Z^{\mathbf N}$ is almost free. What are examples of almost free groups that are ...
3 votes
0 answers
217 views

What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?

Follow up question, edited in on 12/20 below: Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
10 votes
1 answer
544 views

Can automorphism equivalence in a free group be detected in a nilpotent quotient?

If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$. Let $F = F_2$ be the free group on two ...
12 votes
1 answer
510 views

Generators for the first cohomology of free groups

Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...
3 votes
2 answers
280 views

Units of the group algebra of a free group

Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$? In the case of $n=1$, there are only trivial units: $K[F_1]^\...
14 votes
1 answer
980 views

Recognizing free groups

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...
0 votes
0 answers
108 views

Order of elements in amalgamated free products

Reading the book "A Course in the Theory of Groups" by D. J. S. Robinson, I was looking at the proof of 6.4.3 (iii), which states (suppose we are in the case of two groups): if $G_1$ and $...
1 vote
0 answers
125 views

A generalisation of residual finiteness?

A group $\Gamma$ is Residually Finite (RF) if $\forall g \neq e \in \Gamma$ there is a homomorphism $h: \Gamma \to G$ where $G$ is a finite group such that $h(g) \neq e$. Free groups are known to be ...
2 votes
0 answers
120 views

Further questions to limit groups and an article of Fujiwara and Sela

I already have asked a question to the following article: Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, ...
2 votes
0 answers
132 views

Strong converse of Kazhdan's property (T)

In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...
5 votes
1 answer
277 views

Words which are not inverted by any endomorphism

Let $w$ be a word in a free group $F_2$ of two generators $x_1, x_2$ such that there does not exist any endomorphism of free group which takes $w$ to $w^{-1}$. Let $w_1, w_2$ be two words in the same ...
9 votes
1 answer
363 views

Morse theory on outer space via the lengths of finitely many conjugacy classes

Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...
13 votes
0 answers
195 views

Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?

$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
8 votes
1 answer
304 views

If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G?

$\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some ...
2 votes
1 answer
214 views

Projective G-group

Let $G$ be a fixed group. By definition, a $G$-group is a group $X$ with a $G$-action that respects the group operation of $X$. A free $G$-group means a group freely generated by a free $G$-set. A &...
7 votes
2 answers
254 views

Surjections from genus $n$ surface group to free group of rank $n$

Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
0 votes
0 answers
103 views

Fundamental domain of Möbius transformations

In the book Indra's pearls, Möbius transformations are used to construct Kleinian fractals, which are limit sets of a free group generated by two Möbius transformations $a$ and $b$. In the process of ...
5 votes
2 answers
253 views

Stable equivalence of generating sets of a finitely-generated group?

This question came about when I was naively considering to what extent generating sets for finitely-generated groups are unique. Let $G$ be a group and let $\phi_1, \phi_2 : F_k \to G$ be two ...
10 votes
3 answers
775 views

Non-generating sets in a free group

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 $\{a_i^{-1}x_ia_i\}$ does ...
2 votes
1 answer
139 views

Nontrivial abelianization of torsion-free pro-$p$-group which contains a dense free subgroup is infinite?

Let $G$ be a topologically finitely generated pro-$p$-group. Assume that $G$ is torsion-free, it contains a dense free subgroup of infinite rank and the (topological) abelianization $G^{\text{ab}}$ of ...
4 votes
2 answers
519 views

Length of a product of conjugates of an element in a free group

Let $G$ be a free group generated by a set $S$. For $g\in G$, let $l(g)$ be the length of $g$ with respect to $S$. Now for $a\in G$ and $g_1,\dotsc,g_n\in G$, let $$T=g_1^{-1}ag_1g_2^{-1}ag_2\dotsm ...
10 votes
1 answer
330 views

A subgroup of corank 1 in a free group contains a primitive element?

Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq \...
16 votes
1 answer
331 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-...
16 votes
1 answer
638 views

Elements of a free group that can't be inverted by automorphisms

Let $F_n$ be a free group of rank $n$. Say that $w \in F_n$ is non-reversible if there does not exist any $f \in \text{Aut}(F_n)$ such that $f(w) = w^{-1}$. Original Question. Intuitively, I expect ...
17 votes
3 answers
1k views

Examples of locally hyperbolic groups

It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
7 votes
1 answer
227 views

Primitive elements in a free group with trivial projection

For a free group $F$, an element $w$ is primitive if it is part of some free basis for $F$. Let $\pi:F[x_0,x_1,...,x_n]\rightarrow F[x_1,x_2,...,x_n]$ be defined $\pi (x_0)=1$ and $\pi (x_i)=x_i$ for $...
3 votes
1 answer
136 views

Is is true that a proper subword cannot lie in the normal closure of a word?

Let $F$ be a free group and $w\in F$ a cyclically reduced word. Let $v$ be a non-trivial proper subword of $w$. Is it true that $v\notin \langle w^F\rangle$?
10 votes
1 answer
737 views

How can you order a free group?

A left order on a (discrete) group $G$ is a total order on $G$ satisfying $\forall g,h,k \in G: g < h \implies kg < kh$. A right order is defined symmetrically, and a biorder is an order that is ...
3 votes
3 answers
245 views

Perfect group that is split extension of a normal free subgroup of finite index

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect? Thanks @YCor for reformulating the question.
2 votes
1 answer
194 views

Compute reduced word for element in particular free group

Let's consider some particular free group, for instance, the example $G = \langle A, B \rangle < \text{SL}(2, \mathbb{Z})$ in Wikipedia about ping-pong lemma, and some element $g \in G$. Problem #1 ...
5 votes
2 answers
261 views

If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?

I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question. Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying ...
4 votes
3 answers
259 views

Examples of IF-groups

I have seen that several authors say that an infinite group $G$ is an IF-group (or has the IF-property) if every subgroup of infinite index in $G$ is free (for instance, see https://arxiv.org/pdf/1607....
5 votes
2 answers
224 views

Bound on the period of the identity (in a free group) for an automorphism followed by left-multiplication

Let $F$ be a finite-rank free group, $g$ an element of $F$ and $\Phi\colon F \to F$ an automorphism. Consider the dynamical system $\psi_g\colon F \to F$ defined by $x \mapsto g\Phi(x)$. Say that $g$ ...
3 votes
1 answer
219 views

Injectivity of certain homomorphisms on free groups

Consider free groups $F(A)$ and $F(B)$ on finite generating sets $A,B$. Write $A$ and $B$ as the disjoint unions $A=A_1\sqcup A_2$ and $B=B_1\sqcup B_2$. We consider the free groups $F(A_i)$ and $F(...
2 votes
0 answers
123 views

Probability of a finite cylinder set in a free group

Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
9 votes
1 answer
353 views

Finite presentability of semi-direct product of free group and its commutator subgroup

Let $F_n$ be a free group of rank $n \geq 2$. The group $F_n$ acts on its commutator subgroup $[F_n,\, F_n]$ by conjugation. Let $G = [F_n,\, F_n] \rtimes F_n$. It's not hard to see that $G$ is ...
2 votes
0 answers
104 views

understanding the definition of subgroup of the Grothendieck-Teichmuller group

Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
8 votes
0 answers
163 views

Uniform amenability at infinity

Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$ there is a finite subset $F\subset G$ such that $$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
0 votes
1 answer
341 views

Are all Coxeter groups virtually free or virtually surface groups?

From Surface subgroups of Coxeter and Artin groups (Gordon, Long and Reid, 2003) DOI link, we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi_1$ ...
1 vote
1 answer
124 views

Free partners in semi-direct products

Let $G = N \rtimes K$ be a semi-direct product of groups and suppose that $K$ is a finite group. Call the set $\mathcal{F} = \{ \alpha \in G \mid \langle \alpha, K \rangle = \langle \alpha \rangle \...
10 votes
1 answer
541 views

Length of commutators in the free group

Let $G=F_2$ be the free group of rank $2$. Is there a constant $c>0$ such that the word length $|[u,v]|$ of every commutator $[u,v]=uvu^{-1}v^{-1}$ where $u,v\in G$, $|u|,|v|>0$ is at least $c(|...
5 votes
0 answers
112 views

Computability of the "free envelope rank" of an endomorphism of a free group

Let $F$ be a free group freely generated by the finite set $S$ and $\sigma\colon F\to F$ be a group morphism. We define the free envelope rank of $\sigma$, written $r(\sigma)$, as the smallest $k$ for ...
2 votes
2 answers
337 views

Some questions on a paper of Baumslag and Solitar

I was recently trying taking a look at the paper "Some two-generator one-relator non-hopfian groups" by Baumslag and Solitar where they introduce the groups now known as the Baumslag-Solitar ...
7 votes
3 answers
481 views

Membership to double cosets in free groups

Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group? Has this membership problem been implemented in GAP/Magma? More ...
2 votes
1 answer
213 views

Dirichlet region of a free group

Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...
7 votes
1 answer
202 views

Howson property of automorphism group of $F_2$ and of $F_3$

Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) again finitely generated? That is, does $\operatorname{Aut}(F_2)$ (resp. $\...
1 vote
0 answers
43 views

When does bottom stratum of relative train track map give rise to irreducible outer automorphism of free groups

Let $F_n$ be the free group of finite rank $n$ and let $\mathcal{O} \in \text{Out}(F_n)$. Let $\Gamma$ be a finite graph and $f : \Gamma \to \Gamma$ a relative train track representative of $\mathcal{...

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