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visits member for 5 years, 2 months
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Jan
11
comment Two groups that are the automorphism groups of each other
Having said that, Joel David Hamkins does write in a comment 'I believe that no examples are known with period 2 or larger.' But I think he may be talking about 'strong' isomorphism (ie insisting that the isomorphisms are the maps that send elements to inner automorphisms).
Jan
11
comment Two groups that are the automorphism groups of each other
Sorry, I was too hasty. I meant for the finite case with no centre.
Jan
10
comment Two groups that are the automorphism groups of each other
I think this question is answered in the comments to this question: mathoverflow.net/questions/5635/…
Jan
9
comment Is this statement true?
Since this question is not research-level, I'm voting to close.
Jan
7
comment Non-abelian Grothendieck group
Wait, my mistake - apologies.
Jan
7
comment Non-abelian Grothendieck group
The group still seems to be free, since the relations just say that some generator is equal to a product of other relators.
Jan
7
comment Conjugates and infinite index subgroups of free groups
It generalizes to hyperbolic groups and quasiconvex subgroups.
Jan
7
comment Exotic actions of hyperbolic groups
that seems plausible. Of course, your question would then be equivalent to a famous open problem.
Jan
7
comment Conjugates and infinite index subgroups of free groups
@Pablo, a finite Stallings graph represents an infinite-index subgroup iff some vertex has less than maximal degree.
Jan
7
comment Are there quasiconvex normal subgroups?
Of course. But he did it before hyperbolic groups had been invented, and probably phrased his proof differently. I haven't looked at his paper.
Jan
7
comment Are there quasiconvex normal subgroups?
@Igor, he did Fuchsian groups.
Jan
7
comment Exotic actions of hyperbolic groups
Could you give an example of such an action when $G$ is free?
Jan
7
comment Exotic actions of hyperbolic groups
Algebraically, what does the highly transitive hypothesis mean about a point stabilizer?
Jan
7
answered Are there quasiconvex normal subgroups?
Jan
6
comment Growth of the number of generators in hyperbolic groups
@Pablo, all I meant was that, for a fi subgroup H of a 2-dimensional group G, $b_1(H) \geq 1-|G:H|\chi(G)$, by the multiplicativity of Euler characteristic. Thanks for your e-mail - I'll reply soon!
Jan
5
awarded  Enlightened
Jan
5
awarded  Nice Answer
Jan
4
comment Is residual finiteness a property of “many” finitely presented groups?
Pablo, this is also a good question. Do you want to send me an e-mail?
Jan
4
comment Is residual finiteness a property of “many” finitely presented groups?
(But you might like to check out the work of Sapir and his co-authors on random 1-relator groups with more generators.)
Jan
4
comment Is residual finiteness a property of “many” finitely presented groups?
Pablo, I think this is an interesting question. The result I mentioned about 2-generator 1-relator groups is in a paper of Dunfield and (Dylan) Thurston. They prove that a random 2-generator 1-relator group is fg-free-by-cyclic with probability strictly between 0 and 1. In particular, at least some of the time, such groups have zero rank gradient. As far as I know, nothing is known about models with more generators or relations.