In Bers, genus g is used...
UPDATE:
About the word genus, see the comment of Martin Brandenburg, above.
As a complementary information, A.R. Forsyth, Theory of Functions of a Complex Variable, Cambridge, 1918, writes (last paragraph, p.371):
"If the connectivity of a closed surface with a single boundary be 2p+1, the surface is often said to be of genus p"
In the footnote: (genus) Sometimes class. The German word is Geschlecht; French writers use the word genre, and Italians genere.
By the way, in Portuguese, classe or genero.
On p.109, "Laguerre appears to have been the first to discuss the class of transcendental integral functions"
About the letter I think p, in stands for point. In I.M. James, History of Topology, Elsevier, 1999, we can see on pp. 39, last paragraph, "For the sketch of a proof Poincaré collected all types of differential equations on an algebraic curve of given genus p an with given….". On note (38), same page, "p is regular singular point of the differential equation …"