How is the genus of an open Riemann surface $X$ defined? How does it relate to the genus of a compact Riemann surface into which $X$ embeds as open subset? How can the genus of $X$ be computed as an intrinsic invariant?
I read the paper of Pallete and Samanez.
I would be happy to see some examples of open Riemann surfaces of different genus, finite and infinite.
Genus $g(S)$ of any connected surface $S$ is the maximal cardinality of a set $C$ consisting of simple pairwise disjoint loops $L_c$ in $S$ such that the subsurface $$ S \setminus \coprod_{c\in C} L_c $$ is connected. In particular, if $S\subset T$, where $T$ is also a connected surface, then $g(S)\le g(T)$. One can also show that for oriented surfaces, $g(S)$ equals half the rank of the image of $H^1_c(S; {\mathbb Z})$ in $H^1(S, {\mathbb Z})$ under the natural homomorphism. If you want to get an example of an infinite genus open Riemann surface, take an infinite connected sum of tori. If you want to get an interesting example of zero genus, take ${\mathbb C}$ minus the Cantor set.