Timeline for Monoids and groups of fractions
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 30, 2013 at 16:34 | comment | added | Benjamin Steinberg | Most likely your problem is that modules in which M act noninvertibly are not seen by the classifying space. | |
| Oct 30, 2013 at 16:25 | comment | added | Victor | @BenjaminSteinberg, thanks! That's fantastic, I guess we were there and missed absolutely everything with Vesna :-) Let me digest what you said and hopefully some positive thing can be found how to calculate the number of ends for f.g. Ore's semigroups -- since we kind of cast too much "lament" in that paper that nothing works. | |
| Oct 30, 2013 at 16:18 | comment | added | Benjamin Steinberg | @Victor, computationally it is often easier to compute the cohomology for the group using the monoid because the monoid is smaller. | |
| Oct 30, 2013 at 16:16 | comment | added | Benjamin Steinberg | @Victor, for an Ore monoid $M$ with universal group $G$ one has that the cohomology of M with coefficients in a $G$-module $A$ is the same as the cohomology of $G$ with coefficients in $A$. This is because the natural map $BM\to BG$ of classifying spaces is a homotopy equivalence and cohomology with $G$-module coefficients is cohomology of the classifying space with coefficients in a local system. | |
| Oct 30, 2013 at 16:15 | comment | added | Victor | @BenjaminSteinberg: we used monoid ring over Z_2. But what you suggest is great! Let me check it | |
| Oct 30, 2013 at 16:10 | comment | added | Victor | In general, it seems that geometry of Cayley graphs of semigroups is quite "bad notion", though of course many things are known to be nice in some good cases. May be dynamics of endomorphisms on Cayley graphs would yield something -- we studied recently with Alan Cain growths of endomorphisms of f.g. semigroups -- all is very nice, except that it is not clear why really this is important. And also, talking about group-embeddable semigroups, again no relation between growth of endo on the semigroups and growth of lift of the endo to the universal group -- because of Baumslag-Solitar, of course | |
| Oct 30, 2013 at 16:07 | comment | added | Benjamin Steinberg | @Victor, For Z/2 with constant coefficients it seems to me you should get the same answer because the classifying spaces are homotopy equivalent. Are you using constant coefficients or monoid ring over Z_2? | |
| Oct 30, 2013 at 16:02 | comment | added | Victor | @BenjaminSteinberg: yes, for Graside monoids everything is good. But because homologies over $\mathbb{Z}_2$ for Ore's monoids and their universal groups -- even for $(\mathbb{N},+)$ -- are different in general, it makes no hope that ends for f.g. Ore's mononids could be calculated by analogue of Specker's formula for groups. We write about it at the end in arxiv.org/abs/1302.3500 with Vesna Kilibarda | |
| Oct 30, 2013 at 14:53 | comment | added | Benjamin Steinberg | @Victor, Ore monoids are important in Garside theory. The homology of an Ore monoid and its universal group are the same for coefficients in a ZG module. | |
| Oct 30, 2013 at 14:19 | comment | added | Victor | Sorry again, i ate biscuits so cannot stop: it seems would be really nice to understand what are the group-embeddable semigroups whose universal groups are one-relator. At least what are those with Ore's condition. You see, i'm trying to crystalise "importance" of group-embeddable semigroups, which so far only show that things are wild. | |
| Oct 30, 2013 at 13:48 | comment | added | Victor | Sorry, i recalled now: you may check with Simon Craik that (f.g.) Ore's monoids sometimes may be nice -- say they admit only 1,2 or continuum many ends, but i think it is still unknown whether Stallings theorem for them could be worked out (by ends i mean the more natural undirected ends rather than care about artficial in semigroups setting directions in Cayley graphs) | |
| Oct 30, 2013 at 13:40 | comment | added | Victor | Does anybody know any instances in the past when Ore's monoids would appear naturally in order to do something good. I know only of Grigorchuk's proof that f.g. cancellative semigroups with polynomial growth are exactly virtually nilpotent f.g. cancellative semigroups? | |
| Oct 30, 2013 at 13:14 | comment | added | Victor | Excuse me to mention also this: Baumslag-Solitar monoids $\langle a,b:ab=ba^k\rangle$ have Ore's condition, but how much different is the behaviour of the BS-monoid an dthe corresponding BS-group! In general it seems very little in common between a group-embeddable semigroup and its universal group. | |
| Oct 30, 2013 at 13:06 | comment | added | Victor | From the work of Patrick Dehornoy "Complete positive grop presentations" it follows nicely, using rewriting systems (brilliant tool indeed!), that the group of quotients for Ore's monoids is universal. You may also enjoy Example 23 from a recent preprint by George Bergman arxiv.org/abs/1309.0564, as well as the whole paper | |
| Oct 29, 2013 at 18:24 | vote | accept | Stefan Witzel | ||
| Oct 29, 2013 at 17:46 | vote | accept | Stefan Witzel | ||
| Oct 29, 2013 at 18:22 | |||||
| Oct 29, 2013 at 17:34 | answer | added | Benjamin Steinberg | timeline score: 7 | |
| Oct 29, 2013 at 17:30 | history | asked | Stefan Witzel | CC BY-SA 3.0 |