Genus of the Cartesian product of triangles (Q888606)

Summary: We investigate the orientable genus of \(G_n\), the Cartesian product of \(n\) triangles, with a particular attention paid to the two smallest unsolved cases \(n=4\) and 5. Using a lifting method we present a general construction of a low-genus embedding of \(G_n\) using a low-genus embedding of \(G_{n-1}\). Combining this method with a computer search and a careful analysis of face structure we show that \(30\leq \gamma(G_4) \leq 37\) and \(133 \leq\gamma(G_5) \leq 190\). Moreover, our computer search resulted in more than \(1300\) non-isomorphic minimum-genus embeddings of \(G_3\). We also introduce genus range of a group and (strong) symmetric genus range of a Cayley graph and of a group. The (strong) symmetric genus range of irredundant Cayley graphs of \(\mathbb Z_p^n\) is calculated for all odd primes \(p\).