Questions tagged [free-groups]

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Number of trivializations of a trivial word in the free group

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 $\pi$ ...
მამუკა ჯიბლაძე's user avatar
4 votes
2 answers
829 views

Schreier's index formula

A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...
Pablo's user avatar
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23 votes
1 answer
1k views

products of conjugates in free groups

While trying to carry out some technical arguments in free groups, I have encountered the following problem, to which I don't know the answer. Let $F$ be a free group and let $g,a_1,\ldots,a_n \in F$....
Derek Holt's user avatar
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20 votes
0 answers
577 views

Infinitely generated non-free group with all proper subgroups free

Is there any example of group $G$ satisfying the following properties? $G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group. $H< G$ implies that $H$ is ...
W4cc0's user avatar
  • 599
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 (...
Jean Charles's user avatar
10 votes
1 answer
301 views

Is there a non-free group $G$ whose subgroups are all freely decomposable?

Suppose that $G$ is a group such that every subgroup $H \subseteq G$ (including $G$ itself) is either free or a non-trivial free product, i.e. $H = H_1 * H_2$ with $H_1, H_2$ both non-trivial. Is ...
user32157's user avatar
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10 votes
1 answer
740 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 ...
Ville Salo's user avatar
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9 votes
3 answers
1k views

First-order axiomatization of free groups

Is there a way to axiomatize [non-abelian] free groups in first-order logic using the language of groups (which contains the binary operation symbol $\cdot$, and the constant symbol $e$)? Is there ...
Asaf Karagila's user avatar
  • 38.1k
9 votes
1 answer
516 views

Is a free group a product of f.g subgroups of infinite index?

Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?
Pablo's user avatar
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7 votes
3 answers
482 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 ...
Ashot Minasyan's user avatar
6 votes
0 answers
243 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) \...
Pablo's user avatar
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6 votes
1 answer
422 views

Do free profinite groups satisfy Howson's theorem?

Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
Pablo's user avatar
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5 votes
1 answer
279 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 ...
Shri's user avatar
  • 273
5 votes
1 answer
268 views

Nielsen-Schreier with operations

The Nielsen-Schreier theorem states that subgroups of a free subgroup are free. Is this hold also for groups with operations? Explicitly, let $G$ be a fixed group. Let $F$ be a group with $G$-action ...
user49822's user avatar
  • 2,033
5 votes
1 answer
199 views

Converse of Schreier theorem

I know that every subgroup of a free group is free (Schreier theorem). I'm wondering that a (non-trivial) converse is true, that is, if every proper subgroup of an infinite group $G$ is free, then $G$ ...
Wonyong Jang's user avatar
4 votes
0 answers
258 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 ...
Pablo's user avatar
  • 11.2k
4 votes
1 answer
466 views

Generating infinite index subgroups of a free group

Let $F$ be nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index and let $x,y \in F \setminus H$. Must there be some $a \in F$ such that $[F : \...
Pablo's user avatar
  • 11.2k
3 votes
0 answers
404 views

Marshall Hall's theorem for surface groups [closed]

Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$ Let $H \leq \Gamma_g$ ...
Pablo's user avatar
  • 11.2k
3 votes
1 answer
663 views

Bases of free groups

Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...
Pablo's user avatar
  • 11.2k
2 votes
0 answers
142 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 ...
Pablo's user avatar
  • 11.2k
0 votes
0 answers
120 views

Intersection of subgroup of a free group with the lower central series

If I have a subgroup $S$ of a free group $\mathcal{F}_m$, what can I say about the behaviour of the descending sequence of subgroups $\left< S, \Gamma_c(\mathcal{F}_m) \right>$ (where $\Gamma_c(\...
Thomas Meyer's user avatar