Questions tagged [free-groups]
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21 questions
13
votes
1
answer
543
views
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 $\pi$ ...
- 18.4k
4
votes
2
answers
861
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 ...
- 11.3k
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 $g,a_1,\ldots,a_n \in F$....
- 37.4k
20
votes
0
answers
625
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 ...
- 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 (...
- 383
10
votes
1
answer
340
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 ...
- 337
10
votes
1
answer
821
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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 ...
- 6,652
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 ...
- 39.7k
9
votes
1
answer
521
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 $HK = F$ ?
- 11.3k
7
votes
3
answers
523
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 ...
- 3,181
6
votes
0
answers
245
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) \...
- 11.3k
6
votes
1
answer
431
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?
- 11.3k
5
votes
1
answer
294
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 ...
- 355
5
votes
1
answer
232
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 $G$ ...
- 125
5
votes
1
answer
272
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 ...
- 2,178
4
votes
1
answer
515
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 : \...
- 11.3k
4
votes
0
answers
259
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 ...
- 11.3k
3
votes
0
answers
421
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 $H \leq \Gamma_g$ ...
- 11.3k
3
votes
1
answer
739
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 $H$ ...
- 11.3k
2
votes
0
answers
147
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 ...
- 11.3k
0
votes
0
answers
132
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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(\...
- 31