turkmath.org

In 2017 Ghaswala and Winarski classified finite cyclic regular branched coverings of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface, answering a question of Birman and Hilden. In this talk, we will present generalizations of  this result in two directions. First, we will replace finite cyclic groups with finite abelian p-groups. Second, we will replace the base surface with the real projective plane. The main tool is the algebraic characterization of such coverings in terms of the automorphism groups of these finite abelian p-groups. Due to computational insufficiencies we have complete results only for groups of rank 1 and 2. In particular, we prove that for a regular branched $A$-covering $\pi:\Sigma\rightarrow S^2$, where $A={\mathbb Z}_{p^r}\times{\mathbb Z}_{p^t}, \ 1\leq r\leq t$, all homeomorphisms $f:S^2 \to S^2$ lift to those of $\Sigma$, if and only if $t=r$ or $t=r+1$ and $p=3$. If time permits we will also present some applications to automorphisms of Riemann surfaces. This talk mainly is based our work joint with F. Atalan and E. Medetoğulları.