Timeline for What about a Cayley n-complex for n>2?
Current License: CC BY-SA 4.0
25 events
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| May 13, 2022 at 4:18 | comment | added | Sebastien Palcoux | @IJL You should keep your first comment if you think it is still relevant. | |
| May 11, 2022 at 14:36 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
minor edit (title, shape)
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| May 9, 2022 at 15:27 | comment | added | Sebastien Palcoux | @IJL: a $3$-block can be a non-sphere surface, but then it should not be irreducible, because its $π_1$ is non-trivial and a non-nullhomotopic loop should subdivide it into smaller $3$-blocks (this argument comes from a comment of the mentioned post). Problems should arise for irreducible $n$-blocks with $n>3$, as there are non-sphere $(n−1)$-manifolds with a trivial $π_{n−2}$. So I would be very interested to know whether there exists a finitely presented torsion-free group with a non-sphere irreducible $n$-block, for some $n$. | |
| May 9, 2022 at 8:18 | comment | added | IJL | @SebastienPalcoux: My point was that it is very hard to say what one means by good. I just had a quick look at your idea of irreducible $n$-blocks. In the case $n=1$ it has something in common with Brian Bowditch's idea of a taut loop in the Cayley graph (see his article `Continuously many quasi-isometry classes of 2-generator groups'). For larger $n$ I think there is a problem: why should (for example) a 2-block be a 2-sphere rather than say a 2-torus or some other surface? | |
| May 9, 2022 at 3:00 | comment | added | Sebastien Palcoux | @IJL: Your comment should be relevant, but I am not sure to understand what do you mean by good. The Cayley graph is completely determined by a (say finite) set of generators, and then a (good?) set of relators is given by the set of irreducible loops based on the neutral element $e$ (where irreducible means that it is not the product of two loops of strictly smaller length). I extended this idea inductively here with the notion of irreducible n-block. What do you think? | |
| May 8, 2022 at 20:54 | comment | added | IJL | @SebastienPalcoux: Knowing a good set of generators for a group doesn't help you to find a good set of relators, so why should fixing a group presentation help you to find a good set of attaching maps for 3-cells? | |
| May 8, 2022 at 14:18 | review | Close votes | |||
| May 13, 2022 at 3:09 | |||||
| May 8, 2022 at 14:06 | comment | added | Matt Zaremsky | @SebastienPalcoux I guess I'd say that my intuition is, type $F_n$ shouldn't really help in the quest to actually extend the Cayley 2-complex to a concrete $n$-dimensional thing, either with finitely many orbits of cells, or infinitely many. With type $F_n$ groups, often the most natural complex on which the group acts has infinitely many orbits of cells, and collapsing to finitely many leaves a sort of gross (albeit finite) complex. But this is all just sort of vague intuition. | |
| May 8, 2022 at 12:48 | comment | added | Sebastien Palcoux | @MattZaremsky: I agree that the weak contractibility should be hard. I realized that for the infinite dimensional case, leading me to this question, and your comment makes me realize that it should be hard too in the finite dimensional case, very interesting by the way! Do you think that it would be easier to get such a Cayley $n$-complex for a group of type $F_n$ (finiteness properties of groups), that would be appropriate. | |
| May 8, 2022 at 11:58 | comment | added | Matt Zaremsky | About the linked post: I think this "morally" makes sense, but is weak contractibility really that easy? In the Cayley 2-complex it's not true that every embedding of $S^1$ is homotopic to the border of the realization of an irreducible 2-block, that would be like saying every relator in the group is a conjugate of a defining relator (when really you have to use all products of conjugates of defining relators). Similar issues come up for higher $\pi_n$ too. I'm not saying I doubt that this complex is weakly contractible, I'm just saying, I think it's pretty hard. | |
| May 8, 2022 at 11:40 | comment | added | Benjamin Steinberg | I didn't read through the whole linked post. I'm not really a topologist | |
| May 8, 2022 at 4:35 | comment | added | Sebastien Palcoux | @BenjaminSteinberg: what about the construction I mentioned at the end of the post? It should provide the cubical complex you mentioned on the right-angled Artin groups. But it should work for a much larger class of groups (I guess). I wonder what it provides on the Stallings example: the first part (about irreducible $n$-blocks) should be fine (as it should be well-defined in general), but problems may occur in the second part (about geometric realization). Anyway, the first part can still be interesting/useful as a "proto"-CW-complex, a kind of "combinatorial" total space, isn't it? | |
| May 7, 2022 at 20:46 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
clarification: the question is about a natural construction given by any finite presentation
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| May 7, 2022 at 20:28 | comment | added | Benjamin Steinberg | You might look at mathoverflow.net/questions/15957/… | |
| May 7, 2022 at 20:25 | comment | added | Benjamin Steinberg | Also it is undecidable is a finitely presented group has a BG with a finite 3-skeleton which again suggests something nice is hard to find | |
| May 7, 2022 at 20:18 | comment | added | Benjamin Steinberg | Stallings paper is jstor.org/stable/2373106. Bieri later gave similar examples where the n-skeleton of BG is finite but not the n+1 skeleton and Bestvina-Brady groups show things are very messy. So it seems hard to believe a finite presentation can lead to anything ask that nice. | |
| May 7, 2022 at 20:15 | comment | added | Benjamin Steinberg | Yes it is a subgroup of a direct product of free groups. | |
| May 7, 2022 at 20:08 | comment | added | Sebastien Palcoux | @BenjaminSteinberg Do you see any obstruction preventing the existence of a general approach for constructing a model of $EG$ from a finite presentation? Do you have a reference for that example of Stallings? Is it torsion-free? | |
| May 7, 2022 at 19:51 | comment | added | Benjamin Steinberg | @Carl-FredrikNybergBrodda the thing the OP wants is the universal cover of the Salvetti complex. And it can be described easily as filling in the 1-skeleton of any k-cube in the Cayley graph by an actual cube. So this is the kind of thing the OP wants but it is special to this class. | |
| May 7, 2022 at 19:49 | comment | added | Benjamin Steinberg | I don't believe any general approach is known for constructing a model of EG from a finite presentation and Stallings for example early on have a finitely presented group with no model of BG having a finite 3-skeleton. | |
| May 7, 2022 at 19:46 | comment | added | Sebastien Palcoux | @IJL Yes, but if $G$ is a finitely presented group, do you know a (natural) model for $EG$ given by the finite presentation (and such that the $1$-skeleton is the Cayley graph of this presentation, and the $2$-skeleton its Cayley complex)? | |
| May 7, 2022 at 17:04 | comment | added | Carl-Fredrik Nyberg Brodda | The construction for right-angled Artin groups mentioned by @BenjaminSteinberg is called the Salvetti complex, for reference. | |
| May 7, 2022 at 16:39 | comment | added | Benjamin Steinberg | You can always use a bar construction of you want something that works generically. Some classes of groups have nice constructions like right angled artin groups,and one relator groups. | |
| May 7, 2022 at 15:16 | comment | added | IJL | Isn't what you are looking for just the $n$-skeleton of a model for $EG$? | |
| May 7, 2022 at 15:07 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |