Action of Automorphism Group on Homology Group How the group of automorphisms of a compact Riemann surface acts on the first homology group of the Riemann surface?
[Here, by an automorphism of a Riemann surface, we mean conformal self homeomorphism of the Riemann surface. In the paper "Action of automorphisms of a Compact Riemann surface on the first homology group", the author A. M. Macbeath had mentioned the statement (in my question) directly. I want to see the action.]
 A: This will address the OP's question and some comments by niyazi above.  There is action on singular homology by a group of topological maps: given a singular simplex $\sigma: \Delta^n \to X$ and given a continuous map $f: X \to X$, we can consider the singular simplex $f \circ \sigma : \Delta^n \to X$.  This induces a well-defined map on the $n$-th singular homology group of $X$.  In fact if $f$ is homotopic to $g: X \to X$, then $f$ and $g$ induce the same action.
As to studying the action of topological maps, this group is too big.  You could study the group of homeomorphisms of a closed orientable surface up to homotopy; this is the mapping class group; it is finitely generated and has a very nice set of generators (Dehn twists).  And the action of these generators on 1st-homology is easily computed if a nicely compatible basis for homology is chosen.
The OP is looking at an article where there is a given conformal structure and the there is a group of automorphisms of the surface with this structure.  This is a finite group or order less than or equal to $84(g-1)$, where g is the genus of the closed orientable surface $X$.
So, to "see" the action on 1-st homology, you need to specify one of these groups.  You can consider the action in certain cases.  For instance, you can study the action of the group of conformal automorphisms of order 168 on the genus 3 surface realized as the quotient of the $(2,3,7)$ triangle group acting on the hyperbolic plane.  Already, this seems like a challenging problem.  Rather than aim for an explicit linear representation of this group, you might look for something less ambitious.  In MacBeath's article, he does just this and is able to compute the characters of the linear representations.
I hope that the above helps to clarify the question; perhaps some other readers will have some ideas.  Good luck!
