The automorphisms of free groups with boundaries form a family of groups closely related to mapping class groups, with the standard automorphisms of free groups as and (essentially) the symmetric automorphisms of free groups as . We construct a contractible space on which acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of . We also give a presentation of the groups and calculate their first homology group.
Craig Jensen. Nathalie Wahl. "Automorphisms of free groups with boundaries." Algebr. Geom. Topol. 4 (1) 543 - 569, 2004. https://doi.org/10.2140/agt.2004.4.543