A Riemann-Hurwitz Formula for Skeleta in Non-Archimedean Geometry

Let $\varphi :C^{\mathrm{\prime }}\to C$ be a finite morphism between smooth, projective, irreducible curves defined over a non-archimedean valued, algebraically closed field $k$. This morphism induces a morphism between the analytifications of the curves. We will construct a compatible pair of deformation retractions of $C^{\mathrm{\prime }an}$ and $C^{an}$ whose images ${\mathrm{{\rm Y}}}_{C^{\mathrm{\prime }an}}$ and ${\mathrm{{\rm Y}}}_{C^{an}}$ are closed subspaces of $C^{\mathrm{\prime }an}$ and $C^{an}$ which are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces $C^{\mathrm{\prime }an}$ and $C^{an}$ are such that their complements in the two analytifications decompose into the disjoint union of Berkovich open balls and annuli. To these skeleta we can associate a genus. The pair of compatible deformation retractions forces the morphism ${\varphi }^{an}$ to restrict to a map ${\mathrm{{\rm Y}}}_{C^{\mathrm{\prime }an}}\to {\mathrm{{\rm Y}}}_{C^{an}}$. We will study how the genus of ${\mathrm{{\rm Y}}}_{C^{\mathrm{\prime }an}}$ can be calculated using the morphism ${\varphi }^{an}:{\mathrm{{\rm Y}}}_{C^{\mathrm{\prime }an}}\to {\mathrm{{\rm Y}}}_{C^{an}}$.