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bio website dpmms.cam.ac.uk/~hjrw2
location Cambridge
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visits member for 5 years, 4 months
seen 8 hours ago

Mar
28
comment Obtain any 3-manifold from repeating surgeries on knots in $S^3$
@SamNead, I take your point about the base case. As for the other components of the argument - sure, they were non-trivial to spot at the time, but they're fairly standard now (as is true of many important theorems). I was really trying to address Igor's query about the gap between getting infinitely many generators and finitely many.
Mar
25
comment Decidable properties of the Cayley complex of a presentation
The list of possible 'geometric properties' is so long that this question might go on forever. But it's certainly the case that you can decide a lot of things by looking at the link $L(P)$ of the unique vertex of the presentation complex of $P$. For instance, $X(P)$ should be planar (ie embeddable in $S^2$) if and only if the cone on $L(P)$ is planar.
Mar
25
comment Normal subgroup of a totally ordered group
@YCor, yes, in the light of Dave's answer I realize that now.
Mar
23
comment Obtain any 3-manifold from repeating surgeries on knots in $S^3$
Isn't the difference between being generated by Dehn twists and being generated by finitely many Dehn twists essentially just the (obvious) statement that the action of the mapping class group on the curve complex is cocompact?
Mar
16
comment Parodies of abstruse mathematical writing
This is highly reminiscent of the exams from 1066 and all that (in another discipline).
Mar
15
comment Is there a non right-orderable torsion-free factor of the Braid group on 3 strands?
What do you mean by a factor?
Mar
15
comment uniqueness of quotients of principal congruence subgroups
So then I don't see how what you want can possibly be true. Any 2-generated finite group $Q$ will have many epimorphisms $\Gamma(2)\to Q$ (one for each generating pair), and the kernels will be different unless they differ by an automorphism of $Q$. (Unless the groups $\Gamma(2)/\Gamma(2^n)$ have the very special property that every generating pair differs by an automorphism.)
Mar
14
comment uniqueness of quotients of principal congruence subgroups
Isn't $\Gamma(2)$ (in $PSL_2(\mathbb{Z})$) a free group?
Mar
14
comment How does one justify funding for mathematics research?
One problem with this answer is that it doesn't address the point made by Campello, that mathematics funding in total accounts for a relatively tiny fraction of all science funding. Of course the odds of winning are long, but the amount being bet on a win is also pretty tiny. The examples of Euler and Gauss aren't really relevant either. Surely Turing and von Neumann are more pertinent examples.
Mar
13
comment Are all free groups linear, i.e., admit a faithful representation to GL(n,K) for some field K ?
@YCor, sorry, yes, I meant `fully residually linear of dimensions $d$'.
Mar
13
comment Are all free groups linear, i.e., admit a faithful representation to GL(n,K) for some field K ?
The fact that 'locally fully residually linear' implies linear was noticed by Tarski.
Mar
13
comment Decision problem on triviality of intersection of two subgroups
Ditto.${}{}{}{}$
Mar
12
comment Use of an appendix in a long paper
Since you don't seem to have a concern about option 2, why not go with that!?
Mar
9
revised Decision problem on triviality of intersection of two subgroups
Corrected typo.
Mar
9
comment Decision problem on triviality of intersection of two subgroups
You just beat me to it!
Mar
9
comment Decision problem on triviality of intersection of two subgroups
In the negatively curved world (eg hyperbolic groups, free groups etc) the correct hypothesis is that the subgroup should be not just finitely generated but quasiconvex (in some suitable sense). For these subgroups, Stallings' techniques argument in free groups can be generalized (see a recent paper of Kharlampovich--Myasnikov--Weil). The Rips consruction argument I described in my answer shows that some hypothesis along these lines is necessary.
Mar
9
answered Decision problem on triviality of intersection of two subgroups
Mar
5
comment Residual finiteness: why do we care?
Oh, I see. Of course, in that case you already have an explicit and efficient solution to the word problem!
Mar
4
comment Residual finiteness: why do we care?
Out of interest, Igor, do you ever find yourself (in practice) in the situation where you have a finite presentation of a matrix group but no explicit faithful representation?
Mar
3
comment Residual finiteness: why do we care?
I heard a great talk from Colva Roney-Dougal recently (OK, summer 2013, I suppose) about practical algorithms for solving the word problem in negatively curved groups. No doubt she can tell you all about it when you move!