On conformal qc geometry, spherical qc manifolds and convex cocompact subgroups of $$\mathrm{Sp}{(n+1,1)}$$ Sp ( n + 1 , 1 )

Abstract

Conformal qc geometry of spherical qc manifolds is investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. On a scalar positive spherical qc manifold, we can construct the Green function of the qc Yamabe operator, which can be applied to construct a conformally invariant tensor. It becomes a spherical qc metric if the qc positive mass conjecture is true. Conformal qc geometry of spherical qc manifolds can be applied to study convex cocompact subgroups of $$\mathrm{Sp}(n+1,1).$$ On a spherical qc manifold constructed from such a discrete subgroup, we construct a spherical qc metric of Nayatani type. As a corollary, we prove that such a spherical qc manifold is scalar positive, negative or vanishing if and only if the Poincare critical exponent of the discrete subgroup is less than, greater than or equal to $$2n+2$$ , respectively.