All Questions
Tagged with combinatorial-group-theory free-groups
17 questions
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
9
votes
1
answer
390
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 ...
- 93
7
votes
1
answer
219
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. $\...
0
votes
0
answers
132
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(\...
- 31
5
votes
0
answers
192
views
Description of quasimorphisms of the free group
Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...
- 435
13
votes
1
answer
1k
views
Is there a name of semidirect product of a group with its automorphism group?
Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...
- 885
6
votes
1
answer
405
views
An algorithm determining whether two subgroups of a finitely generated free group are automorphic
In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two ...
- 193
12
votes
1
answer
415
views
"Bisecting" a free subgroup with respect to word length
My broad question is regarding the lengths of (reduced) words in a subgroup of a free group.
As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...
- 949
3
votes
1
answer
133
views
Maximal power in a sequence of iterated commutators in the rank two free group
I have the following problem: in the free group $F_2=\langle a,b\rangle$, we define the sequence
$\begin{cases}
w_0=a, \\
w_1=b, \\
w_{n+2}=[w_{n+1},w_{n}] & \text{for }n\ge 0.
\end{cases}$
So $...
- 1,317
2
votes
0
answers
149
views
Concentration of Reduced words
This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback.
...
- 949
3
votes
2
answers
700
views
Subgroup of a free group that is characteristic but not totally characteristic
Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...
- 193
3
votes
2
answers
287
views
Free subgroup of a quotient
Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of cyclically reduced words $w$ in $F$ where the letter $z$ appears at ...
- 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
4
votes
2
answers
337
views
A Karrass-Solitar theorem for surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...
- 11.3k
5
votes
1
answer
264
views
Bases of surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...
- 11.3k
9
votes
2
answers
3k
views
Number of subgroups of a given index of a free group
Given $n,d\in \mathbb{Z}^+$, how many subgroups of index $d$ does the free group of
rank $n$ have?
In case $n=1$ the question is trivial, and in case $n=2, d=2$ there are 3 such subgroups.
I think I ...
- 333
16
votes
1
answer
2k
views
"Concretely" writing down elements in a free profinite group
Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...
- 25.6k