Incidence complexes realization and profinite Rigidity

Abstract

We prove realization theorems for marked profinite groups by means of divisible fillings and completed cellular-incidence complexes. For a nonseparating curve on a closed surface, the divisible curve-power quotients realize the closure of the cut-surface subgroup as an exact intersection and determine the full profinite cut tree. For a nonseparating filling pair, the completed dual square complex is the closed crossing subcomplex of the product of the two cut trees. Its incidence maps realize the given discrete surface action up to a single profinite translation. The same argument applies to free cocompact actions on connected, locally finite, finite-dimensional regular CW complexes. A graph version permits finite vertex stabilizers and assumes equivariant incidence isomorphisms only at cofinally many characteristic finite quotients. The finite-intersection property supplies compatible maps, and the same construction identifies the outer automorphism groups of the discrete and completed marked actions. Full simplex-face incidence yields PL realization of finite simplicial complexes. We apply these results to Kleinian groups. In the lattice case, an integral cohomological comparison, Massey triple products, and an exact correspondence of prime periodic orbits produce the required surface markings, while unit-invariant divisible orbifold fillings treat the cusped case. For arbitrary finitely generated Kleinian groups, equivariant graph markings give boundary-subgroup correspondence and PL realization of marked compact cores. A Schottky example shows that such markings cannot in general be omitted outside the lattice setting.

56 pages