bio  website  dpmms.cam.ac.uk/~hjrw2 

location  Cambridge  
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visits  member for  5 years, 6 months 
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1d

comment 
What exactly is wrong with this statement (LucasPenrose 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 'hprinciple', 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.grouptheory tag. 
May 20 
revised 
Which Lfunctions are not “LanglandsShahidi Lfunctions”?
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 coHopfian, I assume you're not just talking about the trivial examples $\mathbb{Z}^n$? What's a nontrivial 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 3manifolds' for a comprehensive discussion of 2dimensional 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 torsionfree) 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 LouderTouikan 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 DelzantPotyagailo result. It relies on their Topology paper proving 'hierarchical accessibility', which contains an error. This has been fixed by LouderTouikan (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 coHopfian, 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 coHopfian  free groups aren't, for instance. Sela showed that all freely indecomposable hyperbolic groups are coHopfian, and his work almost certainly extends to the toral relatively hyperbolic case (which includes all Kleinian groups). 
May 5 
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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 fixedpoint free action extends to a fixedpoint 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! 