All Questions
12 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
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
11
votes
2
answers
778
views
History of Tarski's problems on free groups
As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...
- 893
11
votes
0
answers
382
views
Ascending chain condition for 1-element normal closures in a free group
Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?
In other words, can there exist elements $...
- 3,181
10
votes
1
answer
534
views
The Tits alternative for $\operatorname{Out}(F_n)$
Not sure if this is the right place to ask this, but the paper I am reading seems to be too specialised for mathstack (if you do not agree, pleas let me know and I will take down this question)
I am ...
- 275
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In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two ...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
4
votes
3
answers
320
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....
- 11.3k
3
votes
0
answers
209
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 ...
- 11.3k
2
votes
2
answers
1k
views
Magnus' embedding theorem
I am looking for a (preferably modern) reference to the following old result of Magnus.
Let $F$ be a free group of finite rank and
$$ F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots ...
- 5,050
2
votes
1
answer
153
views
Collections in direct products and freeness
I am looking for references about the following type of questions:
Let $G$ and $H$ be two groups,
let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements,
and ...
user34424
2
votes
0
answers
114
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 ...
- 204
0
votes
0
answers
105
views
specific qi on free groups
Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...
- 1,528