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The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar connected Cayley graph). Results of Tucker (https://doi.org/10.1016/0095-8956(84)90031-5) imply that there are only finitely many groups $G$ of non-orientable genus $\overline{\gamma}(G)=k$ for every $k>3$. Clearly, there are infinitely many groups with $\overline{\gamma}(G)=0$. Moreover, using results of Riskin (https://www.sciencedirect.com/science/article/pii/S0012365X0000193X) one can show that $\overline{\gamma}(\mathbb{Z}_p\times \mathbb{Z}_{p^k})=3$ for every prime $p\geq 5$ and positive integer $k$. I wonder:

Question 1: Are there infinitely many groups of non-orientable genus $1$, i.e., they have a connected Cayley graph embeddable in the projective plane, but no planar one?

Question 2: Are there infinitely many groups of non-orientable genus $2$, i.e., they have a connected Cayley graph embeddable in the Klein bottle, but none embedabble in the projective plane?

Questions 3: Is there any group of non-orientable genus $2$? ($\mathbb{Z}_3\times\mathbb{Z}_3$ has non-orientable genus $1$)

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    $\begingroup$ 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. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Aug 16, 2022 at 12:44
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    $\begingroup$ Right, however I only know of one such group $\mathbb{Z}_3\times\mathbb{Z}_3$ for sure. $\endgroup$
    – Kolja Knauer
    Aug 16, 2022 at 13:01
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    $\begingroup$ And how about $\mathbb{Z}_3 \times \mathbb{Z}_n$ for $n>3$? Is this genus $1$? $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Aug 16, 2022 at 13:34
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    $\begingroup$ 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/… $\endgroup$
    – Kolja Knauer
    Aug 16, 2022 at 14:33
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    $\begingroup$ 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 $\endgroup$
    – Kolja Knauer
    Aug 16, 2022 at 14:45

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