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Abstract

A compact Riemann surface $X$ of genus $g>1$ is said to be $p$-hyperelliptic if $X$ admits a conformal involution $\varrho$, called a $p$-hyperelliptic involution, for which $X/\varrho$ is an orbifold of genus $p$. If in addition $X$ admits a $q$-hypereliptic involution then we say that $X$ is $pq$-hyperelliptic. We give a necessary and sufficient condition on $p,q$ and $g$ for existence of a $pq$-hyperelliptic Riemann surface of genus $g$. Moreover we give some conditions under which $p$- and $q$-hyperelliptic involutions of a $pq$-hyperelliptic Riemann surface commute or are unique.