Nilpotent automorphism groups of bordered Klein surfaces

Abstract:

Let $X$ be a compact bordered Klein surface of algebraic genus $g \geqslant 2$, and let $G$ be a group of automorphisms of $X$. Then the order of $G$ is at most $12(g - 1)$. Here we improve this general bound in an important special case. We show that if $G$ is nilpotent, then the order of $G$ is at most $8(g - 1)$. This bound is the best possible. We construct infinite families of surfaces that have a nilpotent automorphism group of order $8(g - 1)$. The nilpotent groups of maximum possible order must be $2$-groups. We prove that if the nilpotent group $G$ acts on a bordered surface of genus $g$ such that $o(G) = 8(g - 1)$, then $g - 1$ is a power of 2. Further, our examples show that for each nonnegative integer $t$ there is a bordered surface with genus $g = {2^t} + 1$ and a group of automorphisms of order $8(g - 1)$.