Tucker, Thomas W.; Matematični kolokvij oktober 2006

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Genus Gaps for Maps

Thomas W. Tucker

Colgate University

19. oktober 2006

We consider various questions about the existence of certain kinds of maps on a surface of a given genus. When no such map exists, we say there is a gap at that genus. For example, computer calculations have shown there are many "small" nonorientable surfaces having no regular maps. On the other hand, only very recently were Gardner, Nedela, Širáň, and Škoviera able to show there are infinitely many such gaps, namely for nonorientable genus p + 2, where p = 1 mod 12 and p > 13. Another example is chiral regular maps on orientable surfaces, where recent computer calculations for genus at most 200, show many gaps. It is conjectured that such gaps exist for all genera of the form p + 1, where p is an odd prime such that p − 1 is not divisible by 3, 5, or 8. A third example is for the symmetric genus of a finite group A (the smallest genus surface on which A acts faithfully) or the White genus (the smallest genus surface containing an embedded Cayley graph for A). Many years ago we showed that there is only one group of White genus 2, suggesting there may be gaps at other genera. In joint work with Conder in the last year, we have shown that any gaps for the symmetric genus can occur only for g = 2,8,14 mod 18.

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