bio | website | dpmms.cam.ac.uk/~hjrw2 |
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location | Cambridge | |
age | ||
visits | member for | 4 years, 11 months |
seen | 22 hours ago | |
stats | profile views | 5,614 |
Oct 10 |
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Fundamental group of a closed hyperbolic surface is Gromov hyperbolic
This question is not research-level - any introductory text will contain a proof of this fact. Voting to close. |
Oct 4 |
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Subgroups of one-relator groups
A couple more comments. These results are not embeddability results as asked for in the final paragraph of the question, in the sense that they don't say 'Every group of class X is embeddable into a 1-relator group', but rather 'A random 1-relator group contains a subgroup from class X'. But the proof techniques do generate a very large number of subgroups of this form---a generic 1-relator group contains many 3-manifold groups as subgroups. |
Oct 4 |
answered | Subgroups of one-relator groups |
Oct 2 |
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Detecting invertible elements in group rings by their images for finite quotients of the group
Perhaps you should edit your question to indicate that it's not known to be the case that 'most 3-manifold groups' are residually free nilpotent group, then? |
Oct 1 |
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Detecting invertible elements in group rings by their images for finite quotients of the group
Every hyperbolic 3-manifold has a finite-sheeted covering space whose fundamental group is residually torsion-free nilpotent. Is that the statement you're thinking of? |
Oct 1 |
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Has philosophy ever clarified mathematics?
Surely this should be community wiki? |
Sep 30 |
awarded | Explainer |
Sep 28 |
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Presentation of Homotopy Pure Braid Group?
Let me try to summarize what the Reidemeister--Schreier process tells you. The group $S_n$ acts by outer automorphisms on $P_n$. For each relator of type 3, you need to add its $S_n$-orbit to the set of relators for $\widetilde{P}_n$. |
Sep 28 |
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Presentation of Homotopy Pure Braid Group?
Without doing the computation I can't be sure. But the set of relators you add certainly needs to be invariant under the natural $S_n$-action. It's not clear to me that the set of relators of type 3 is invariant. |
Sep 28 |
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Presentation of Homotopy Pure Braid Group?
I take it that the homotopy pure braid group is the kernel of the natural map to $S_n$? Anyway, the Reidemeister--Schreier process provides an algorithm for computing presentations of finite-index subgroups. |
Sep 26 |
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CAT(0) groups that does not act on CAT(0) cubical complex
Small correction: any uniform lattice (since the action is required to be cocompact). |
Sep 24 |
answered | Ends of quotients of Coxeter Groups |
Sep 22 |
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Images of the fundamental domain of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ whose Euclidean area is large
@Igor, it's just how I'd make a nice poster. I said it doesn't answer the question. |
Sep 22 |
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Images of the fundamental domain of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ whose Euclidean area is large
@IgorRivin - I mean that it's easy to list all the elements in a certain ball in the Cayley graph, and you can then use the corresponding matrices to compute the translates of the fundamental domain. Quasi-isometry doesn't come into it. |
Sep 22 |
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Images of the fundamental domain of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ whose Euclidean area is large
It's not exactly an answer to your question, but isn't the easiest thing just to compute the ball of a certain radius $d$ in the Cayley graph? This would be very easy to programme. Because of the amalgamated free product decomposition of $SL_2(\mathbb{Z})$, there will be no redundancy if you choose the right generating set. |
Sep 22 |
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A Karrass-Solitar theorem for surface groups
Interesting argument! I think you mis-spoke slightly: having one end precisely does not imply being virtually infinite cyclic---infinite cyclic groups have two ends. |
Sep 21 |
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Bases for free pro-p groups
I wrote a paper with Zalesskii called 'Profinite properties of graph manifolds', which appeared in Geom. Dedicata. You could look at the references in that. |
Sep 21 |
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Bases for free pro-p groups
Pablo, I don't know how relevant it is to this specific question, but you should look into the theory of profinite and pro-p trees, as developed by Ribes and Zalesskii. (Not in their published book, but in some papers by them and various co-authors, including me.) This technology makes concrete some of the analogies between profinite (or pro-p) free groups and abstract free groups. |
Sep 20 |
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A bound on the size of the center
Oh, right. I thought it was the centralizer. |
Sep 19 |
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A bound on the size of the center
Why is $C_{F/N}(HN/N)$ normal in $F/N$? |