Timeline for Groups of non-orientable genus 1 and 2
Current License: CC BY-SA 4.0
19 events
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| Oct 21 at 14:12 | comment | added | HJRW | @KoljaKnauer: Well, "regular embeddings" sound like they are precisely the finite subgroups of the respective homeomorphism groups. So they seem much closer to lots of mainstream work in geometric topology, and should be well understood. | |
| Oct 21 at 11:37 | comment | added | Kolja Knauer | @HJRW: Yeah, this is called "regular embedding" and leads to another notion of genus. There is also many papers about it. I am not sure how this notion of genus behvaes with respect to the questions in this post. | |
| Oct 21 at 9:42 | comment | added | HJRW | @KoljaKnauer: Thanks! That's pretty complicated. It's interesting that the planar Cayley graphs can all be embedded in such a way that the action extends to the sphere; whereas that can't be true for this example. | |
| Oct 20 at 19:02 | comment | added | Kolja Knauer | I put a drawing once on the torus for reference and then on RP2 here | |
| Oct 19 at 17:03 | comment | added | HJRW | @KoljaKnauer: I have tried to find an embedding into RP^2, but so far no luck. Could you give me a hint? :) | |
| Oct 18 at 13:31 | comment | added | Kolja Knauer | All inclusion-minimal generating sets are essentially the same (this is a vector space), and the cayley graph is the cartesian product of two 3-cycles. This graph is "well-known" (try!) to be embedabble on the projective plane but not planar. | |
| Oct 18 at 12:10 | comment | added | HJRW | I'd be curious to see a proof/reference for the fact that $\mathbb{Z}/3\times\mathbb{Z}/3$ has genus 1. | |
| Oct 17 at 9:02 | history | edited | Kolja Knauer | CC BY-SA 4.0 |
added 1 character in body
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| Oct 16 at 12:09 | history | edited | Kolja Knauer | CC BY-SA 4.0 |
added 7 characters in body
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| Oct 16 at 12:06 | comment | added | Kolja Knauer | I just realized that any minimal Cayley graph of the Quaternion group is isomorphic to $K_{4,4}$, which has non-orientable genus 2. Hence this answers posiitively (former) Question 3, which I will remove now. | |
| Aug 23, 2022 at 16:07 | history | edited | Kolja Knauer |
edited tags
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| Aug 19, 2022 at 9:51 | history | edited | Kolja Knauer | CC BY-SA 4.0 |
corrected/added the statement about $\overline{\gamma}(G)=3$ and added Question 3
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| Aug 16, 2022 at 14:45 | comment | added | Kolja Knauer | Furthermore Corollary 3.11 in sciencedirect.com/science/article/pii/S0012365X0000193X claims that starting from $\mathrm{Cay}(\mathbb{Z}_3\times\mathbb{Z}_5,\{(0,1),(1,0)\})$ one cannot even embed in the Klein bottle | |
| Aug 16, 2022 at 14:33 | comment | added | Kolja Knauer | I do not know wrt other generating systems, but for $m\geq 3, n>3$ the graph $\mathrm{Cay}(\mathbb{Z}_m\times\mathbb{Z}_n,\{(0,1),(1,0)\})$ contains as minor a cube graph $Q_3$ with a vertex adjacent to all members of one bipartition class of $Q_3$. This is an exluded minor for embeddability in the projective plane, see e.g. Figure 1.1. in smartech.gatech.edu/bitstream/handle/1853/45914/… | |
| Aug 16, 2022 at 13:34 | comment | added | Carl-Fredrik Nyberg Brodda | And how about $\mathbb{Z}_3 \times \mathbb{Z}_n$ for $n>3$? Is this genus $1$? | |
| Aug 16, 2022 at 13:01 | comment | added | Kolja Knauer | Right, however I only know of one such group $\mathbb{Z}_3\times\mathbb{Z}_3$ for sure. | |
| Aug 16, 2022 at 12:44 | comment | added | Carl-Fredrik Nyberg Brodda | A relevant answer classifying the genus $0$ case. If one can find a family of groups which have some genus $1$ Cayley graph and which are not in the genus $0$ list, then that answers your first question. | |
| S Aug 16, 2022 at 9:51 | review | First questions | |||
| Aug 16, 2022 at 10:06 | |||||
| S Aug 16, 2022 at 9:51 | history | asked | Kolja Knauer | CC BY-SA 4.0 |