The motivation here is that I'm looking for a group that topologically (based on its Cayley Graph) behaves like a sphere, but algebraically is similar to the direct product of two cyclic groups (which have 4-regular Cayley graphs), but are toroidal, with an Euler characteristic of 0.
2025-01-13 05:19:46.1736745586
Are there any examples of groups with a 4-regular cayley graph with an euler characteristic of 2?
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Yes, there are at least some examples, listed at this really great website http://weddslist.com/groups/cayley-plat/index.html .
In particular,
$A_4$ has a Cayley graph that's the skeleton of the cuboctahedron.
$S_4$ has a Cayley graph that's the skeleton of the rhombicuboctahedron.
$A_5$ has a Cayley graph that's the skeleton of the rhombicosidodecahedron.
The dihedral groups have Cayley graphs that are the skeletons (skeleta? skeleton doesn't look Latin...) of antiprisms. Even better, since you wanted groups "similar to the direct product of two cyclic groups," the dihedral groups are semidirect products $C_n \rtimes C_2$, which is hopefully "similar" enough to a direct product in the sense that you want!
It would be interesting to know what, if any, other examples exist.