Geometry and topology Seminar
Thursday, 6 April, 2023 at 11.30 am
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Venue: Ramanujan Hall
Host: Rekha Santhanam
Speaker: Kashyap Rajeevsarathy
Affiliation: IISER, Bhopal
Title: Generating the liftable mapping class groups of regular cyclic covers
Abstract: Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. We show that the liftable mapping class group $\mathrm{LMod}_k(S_g)$ of the $k$-sheeted regular cyclic cover of $S_g$ is self-normalizing in $\mathrm{Mod}(S_g)$ and that $\mathrm{LMod}_k(S_g)$ is maximal in $\mathrm{Mod}(S_g)$ when $k$ is prime. Moreover, we establish the existence of a normal series of $\mathrm{LMod}_k(S_g)$ that generalizes a well-known normal series of congruence subgroups in $\mathrm{SL}(2,\mathbb{Z})$. Furthermore, we give an explicit finite generating set for $\mathrm{LMod}_k{S_g)$ for $g \geq 3$ and $k \geq 2$, and when $(g,k) = (2,2)$. As an application, we provide a finite generating set for the liftable mapping class group of the infinite-sheeted regular cyclic covering of $S_g$ for $g \geq 3$ by the infinite ladder surface.