In [8] one of the authors constructed uncountable families of groups of type F P and of n-dimensional Poincaré duality groups for each n ≥ 4. We show that the groups constructed in [8] comprise uncountably many quasi-isometry classes. We deduce that for each n ≥ 4 there are uncountably many quasi-isometry classes of acyclic n-manifolds admitting free cocompact properly discontinuous discrete group actions.

We show that for each positive integer k there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as n k . As a consequence we produce examples of right-angled Artin groups containing finitely presented subgroups whose Dehn functions grow as n k`2 .

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus g with b boundary components and n punctures in the cases when {g=0} with b and n arbitrary, and when {g=1} and {b+n} is at most 3, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type {D_{4}}. As a corollary, we conclude that the pure mapping class groups are linear in these cases.

We show that in any right-angled Artin group whose defining graph has chromatic number k, every non-trivial element has stable commutator length at least 1/(6k). Secondly, if the defining graph does not contain triangles, then every non-trivial element has stable commutator length at least 1/20. These results are obtained via an elementary geometric argument based on earlier work of Culler.

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