Convex-compact subgroups of the Goeritz group

Let G > Mod 2 G>\operatorname {Mod}_2 be the Goeritz subgroup of the genus-2 mapping class group. We show that finitely-generated, purely pseudo-Anosov subgroups of G G are convex cocompact in Mod 2 \operatorname {Mod}_2 , addressing a case of a general question of Farb–Mosher. We also give a simple criterion to determine if a Goeritz mapping class is pseudo-Anosov, which we use to give very explicit convex-cocompact subgroups. In our analysis, a central role is played by the primitive disk complex P \mathcal {P} . In particular, we (1) establish a version of the Masur–Minksy distance-formula for P \mathcal {P} , (2) classify subsurfaces X ⊂ S X\subset S that are infinite-diameter holes of P \mathcal {P} , and (3) show that P \mathcal {P} is quasi-isometric to a coned-off Cayley graph for G G .