Weight bounds for $(3,\gamma )$-hyperelliptic curves

{\it $(N,\gamma )$-hyperelliptic} semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated to totally-ramified points of $N$-fold covers of curves of genus $\gamma$. Torres characterized $(2,\gamma )$-hyperelliptic semigroups of maximal weight whenever their genus is large relative to $\gamma$. Here we do the same for $(3,\gamma )$-hyperelliptic semigroups, and we formulate a conjecture about the general case whenever $N\ge 3$ is prime.