  For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we
consider the following type of query: given two vertices $s$ and $t$ in $G$,
what is the weight of a minimum $st$-cut in $G$? We solve this problem in
preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space $O(n)$ that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
$2^{O(g^2)}$.
