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bio website dpmms.cam.ac.uk/~hjrw2
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visits member for 5 years, 6 months
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1d
comment What exactly is wrong with this statement (Lucas-Penrose fallacy)?
This is a famous philosophical fallacy. It's a good question, but not suitable for MO, so I too will vote to close. In case it helps, I seem to remember that the fallacy is discussed in Goedel, Escher, Bach, and also in one of Daniel Dennett's popular books (perhaps Darwin's Dangerous Idea?).
2d
comment John Nash's Mathematical Legacy
@WillieWong, as someone who has often heard the term 'h-principle', but didn't know that it was inspired by Nash (among others), I'm very glad you mentioned it!
May
25
revised Number of trivializations of a trivial word in the free group
Added gr.group-theory tag.
May
20
revised Which L-functions are not “Langlands-Shahidi L-functions”?
Corrected link text.
May
15
comment Splitting over infinite generated abelian subgroup?
Well, for instance, any quasiconvex subgroup of infinite index in a hyperbolic group is contained in an infinitely generated subgroup.
May
14
comment Splitting over infinite generated abelian subgroup?
Re: question 1, let $H$ be a finitely generated CSA group with an infinitely generated abelian subgroup $A$ (I don't know an example of such an $H$, but I'm fairly sure it exists), and let $G=H*\mathbb{Z}$, which is also CSA. Then we also have $G=H*_A (A*\mathbb{Z})$, so $G$ also splits over an infinitely generated abelian subgroup. These kind of 'folding' constructions (the second splitting is a folding of the first) tend to show that most groups that split split over an infinitely generated subgroup.
May
12
comment Continuity of conjugation actions of Polish groups
Just to be clear, am I right in thinking that the issue is that the topoogy on $G$ induced by $\psi$ may not coincide with the intrinsic topology?
May
9
comment Kleinian groups containing an isomorphic copy of itself
@DanielGroves, oh right, of course.
May
8
comment Thickening graphs to get honest actions
Regarding your motivating example, perhaps it's worth pointing out that all automotphisms of $F_2$ are geometric, and can be realized on the punctured torus.
May
8
comment Kleinian groups containing an isomorphic copy of itself
@DanielGroves, good point about the torsion. When you say there are freely indecomposable toral relatively hyperbolic groups that aren't co-Hopfian, I assume you're not just talking about the trivial examples $\mathbb{Z}^n$? What's a non-trivial example?
May
7
comment Is there a simple description of this group?
PS Scott's article can be downloaded from his webpage: math.lsa.umich.edu/~pscott .
May
7
comment Is there a simple description of this group?
Your group is the fundamental group of an orbifold $O$, with genus one and a single cone point of order 2. See Peter Scott's article `The geometries of 3-manifolds' for a comprehensive discussion of 2-dimensional orbifolds. The fact that the (rational) Euler characteristic $\chi(O)=-1/2$ is negative implies that the universal cover of $O$ is the hyperbolic plane, and your group is a cocompact Fuchsian group.
May
6
comment Kleinian groups containing an isomorphic copy of itself
@YCor, you also want Selberg's lemma (to show that they're virtually torsion-free) and the Sphere theorem (to show that the quotient is aspherical). Though perhaps the Sphere theorem is overkill...
May
6
comment Kleinian groups containing an isomorphic copy of itself
I should probably add that the mistake is documented in Example 1 of the Louder--Touikan paper.
May
6
comment Kleinian groups containing an isomorphic copy of itself
Another thing to mention is that there may be a problem with the proof of the Delzant--Potyagailo result. It relies on their Topology paper proving 'hierarchical accessibility', which contains an error. This has been fixed by Louder--Touikan (arxiv.org/abs/1302.5451), but one would need to check that the results needed for the Kleinian groups paper remain true.
May
6
comment Kleinian groups containing an isomorphic copy of itself
For reference, you seem to be asking about cofinitely Hopfian Kleinian groups. As has been pointed out, this property is much weaker than co-Hopfian, and much easier to prove. (Ian's argument above does it.) See arxiv.org/abs/1012.1785v1 and the references therein.
May
6
comment Kleinian groups containing an isomorphic copy of itself
@YCor, there is for arbitary groups virtually of type F, which includes all Kleinian groups.
May
6
comment Kleinian groups containing an isomorphic copy of itself
It's not true that all Kleinian groups are co-Hopfian -- free groups aren't, for instance. Sela showed that all freely indecomposable hyperbolic groups are co-Hopfian, and his work almost certainly extends to the toral relatively hyperbolic case (which includes all Kleinian groups).
May
5
comment Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?
@YCor, to be sure I understand you, when you assert that 'any group of isometries of a subset of a Hilbert space extends to a group of isometries of the whole Hilbert space', you also want that a fixed-point free action extends to a fixed-point free action?
May
5
comment Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?
@DylanThurston, that's why I posted a comment, and not an answer!