  In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics
in which the motion of an inviscid incompressible fluid is described as the
geodesic flow of the right-invariant $L^2$-metric on the group of
volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic,
group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex
sheets. It turns out that the corresponding dynamics is related to a certain
groupoid of pairs of volume-preserving diffeomorphisms with common interface.
We also develop a general framework for Euler-Arnold equations for geodesics on
groupoids equipped with one-sided invariant metrics.
