A group G is hopfian if every onto endomorphism f:G->G is an isomorphism.
We shall describe some extensions of this notion, and examine the case of
semidirect products G=F(r) \rtimes_\phi Z, where \phi is an automorphism of the
free group F(r). (Such groups G are the "mapping tori" of the title.)
Although these notions derive from topology, the arguments are primarily algebraic.
This is based on joint work with M.Bridson, D.Groves and G.Martin.