Arnold's seminar,
Moscow, 24.04.2001

S.M.Natanzon

Topological classification of actions of the group Z^m_p
on surfaces.

By an action of a group G on a surface S we understand
a representation f of the group G in the group of orientation 
preserving autohomeomorphisms of the surface S.
Two actions (S, f) and (S', f') are viewed as equivalent,
if there exists a homeomorphism h: S -> S' such that fh=hf'. 
In this talk, we give a complete description of equivalence 
classes of the actions of the group G=Z^m_p (p is a prime) 
expressed in the terms of the group G itself. In the case of 
actions without fixed points such a class is described by a 
bilinear skew-symmetric form on G with values in Z_p. 
The topological classification allows to describe the connected 
components of the space of complex algebraic curves with the 
group of automorphisms isomorphic to Z^m_p.