Infinite Volume and Bounded Cohomology

The bounded cohomology of a group encodes a wealth of geometric and algebraic data. We will define bounded cohomology of groups and construct explicit examples in dimension three; they come from
computing the volumes of locally geodesic tetrahedra in hyperbolic manifolds. It turns out that these volume classes distinguish the bi-Lipschitz classes of hyperbolic structures of infinite volume on a fixed 3-manifold, a fact that we will use to interpret addition in bounded cohomology as a kind of `geometric connected sum’ on hyperbolic manifolds.