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Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)? The best I could get by trial and error is an embedding ...