We completely classify the locally finite, infinite graphs with pure mapping
class groups admitting a coarsely bounded generating set. We also study
algebraic properties of the pure mapping class group: We establish a semidirect
product decomposition, compute first integral cohomology, and classify when
they satisfy residual finiteness and the Tits alternative. These results
provide a framework and some initial steps towards quasi-isometric and
algebraic rigidity of these groups.Comment: 36 pages, 10 figure