Executive Summary : | The symmetric mapping class group is the group of mapping classes that preserve the fibers of a branched cover between surfaces, while the liftable mapping class group represents mapping classes that lift under the cover. A cover has the Birman-Hilden property if two isotopic fiber-preserving homeomorphisms are isotopic via a fiber-preserving homotopy. The quotient of the symmetric mapping class group by the deck group is isomorphic to the liftable mapping class group. The symmetric mapping class is the normalizer of the deck group of the cover.
Birman-Hilden and Ghaswala-Winarski have both contributed to the field of geometry, with Birman-Hilden's 1971 presentation on hyperelliptic involution and Ghaswala-Winarski's 2017 presentation on the liftable mapping class group of a finite-sheeted balanced superelliptic cover. Recently, a generating set for the liftable mapping class group for an unbranched cyclic cover between closed surfaces was derived.
The authors aim to derive generating sets and presentations for the liftable and symmetric mapping class groups of finite cyclic covers of closed oriented surfaces, derive equivalent conditions for the liftable mapping class group to be a maximal subgroup of the mapping class group of a closed oriented surface, and solve problems 1-2 when the deck transformation group is a finite metacyclic group. They also aim to characterize all periodic mapping classes that lift under finite-sheeted alternating or metacyclic covers of closed oriented surfaces. |