All Questions
10 questions
1
vote
0
answers
162
views
Solution of an equation over free group
Let $F_n$ be a free group on $n$ generators. Let $w \in F_n$ be a word such that there does not exist any solution in $F_n$ for the equation $w.w(t_1, \ldots, t_n) = 1$, where $t_1, \ldots, t_n$ are ...
- 355
33
votes
1
answer
1k
views
Is this conjecture strictly weaker than P=NP?
My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...
user6976
15
votes
2
answers
972
views
Elementary subgroups of surface groups
From Sela's proof of Tarski's conjecture we know that the surface groups (i.e. fundamental group of a closed surface of genus $\geq 2$) and free (non-abelian) groups have the same first order theory. ...
- 1,713
6
votes
1
answer
226
views
Are $G$-limits of a slender group $G$ in the space of marked groups also slender?
A group $G$ is slender if every subgroup $H \leq G$ is finitely generated. This includes polycyclic-by-finite groups. Such groups are also called noetherian.
Suppose that $L$ is a $G$-limit group in ...
- 1,033
2
votes
0
answers
90
views
Fully residually free groups and completion
Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
- 11.3k
3
votes
1
answer
374
views
Is a finitely generated residually free group "almost LERF"?
Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
Let $...
- 11.3k
3
votes
2
answers
290
views
Do limit groups satisfy Howson's theorem?
Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated
subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must
$A \cap B$ be finitely generated?
Recall that a limit ...
- 11.3k
5
votes
1
answer
884
views
solvable word problem without algorithm
Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...
- 829
8
votes
0
answers
298
views
A Magnus theorem in the category of residually finite groups
There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle X\...
- 38.6k
21
votes
4
answers
2k
views
Is there a non-Hopfian lacunary hyperbolic group?
The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...
- 25k