Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-trivial isometry group? 



Isometry group of a compact hyperbolic surface
In genus 2, every surface has a symmetry, namely a hyperelliptic involution. In higher genus, generic surfaces will not have any symmetries. If a surface has a non-trivial symmetry group, then the quotient by the symmetry group will be an orbifold, and the moduli space of hyperbolic structures on this orbifold will have strictly smaller dimension. Hence moduli space will not be covered by these smaller dimensional Teichmüller spaces. 


One possible obstruction: suppose a surface admits a symmetry (not hyperelliptic in genus 2), then there will be simple closed geodesics which get moved by an isometry, and hence there are simple closed curves which have the same lengths. So if a surface has all simple closed curves of different lengths, then it cannot admit any non-trivial symmetries. One could probably also state this for curves of some bounded length depending on the genus and injectivity radius. 


To see that some curve must be sent to a distinct curve, suppose not. Then every non-separating curve intersects another curve in a single point. Then this point must be fixed by the isometry, and in fact the isometry must be an involution fixing this point. Arguing like this, one can conclude that the isometry is a hyperelliptic involution. However, in genus $>2$, there are always curves not fixed by any given hyperelliptic. 


