Questions tagged [one-relator-groups]
Finitely presented groups having a single relator
9 questions
9 votes
1 answer
995 views
What is this theorem in Egorov 1981?
In a paper by Wise it is claimed that: [...] Egorov proved the residual finiteness of positive one-relator groups where the relator is of the form $W^n$ and $n\geq 2$. A paper by Baumslag, Miller, ...
- 491
6 votes
1 answer
237 views
When is the double of a free group (along a cyclic subgroup) isomorphic to a surface group?
Let $F_{2n}$ be the free group of rank $2n$, and let $w\in F_{2n}$ be a word. Let $D=F_{2n}\ast_{\langle w \rangle} F_{2n}$ be the double of $F_{2n}$ along $w$. In other words, suppose $F_{2n}=\langle ...
- 345
0 votes
0 answers
71 views
Are the presentations of free profinite group as an one-relator group essentially unique?
Let $F$ be the free profinite group of rank $r$. Let $\tilde{F}$ be the free profinite group of rank $r + 1$, and $C$ be a subgroup of $\tilde{F}$ which is, of course non-canonically, isomorphic to $\...
- 103
9 votes
1 answer
423 views
Is this cyclically pinched one-relator group fully residually free?
Consider the following one-relator group: $$G = \langle x,y,w,z \mid x^3[x,y][w,z] = 1\rangle$$ where $[a,b] = aba^{-1}b^{-1}$ denotes the commutator. It is the free product with amalgamation $$F(x,y) ...
- 515
9 votes
3 answers
980 views
Is there a one relator group with property (T)?
Is there a one-relator group with property (T)? That is, is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup ...
- 11.4k
11 votes
2 answers
1k views
Subgroups of one-relator groups
I know that not every finitely-presented group may be embedded into a one-relator group, for example because of a theorem of Magnus stating that the word problem is solvable in one-relator groups. But ...
- 2,439
11 votes
1 answer
542 views
1-related groups
Every 1-related group $G$ with at least 2 generators is an HNN extension of another 1-related group $G_1$ with free associated subgroups. Indeed, if the total exponent of one letter in the relator is ...
user6976
10 votes
2 answers
780 views
Freiheitssatz implies a finitely generated one relator group embeds in a two-generator one relator group?
I've read that every finitely generated one relator group embeds in a two generator one relator group, and that this fact follows from the Freiheitssatz. Unfortunately, the only proof I can find of ...
- 736
24 votes
4 answers
2k views
Free splittings of one-relator groups
Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is it ...
- 26.1k