Croke–Kleiner Admissible Groups: Property (QT) and Quasiconvexity

K-quasiconvexity Reduces to Quasiconvexity

The relation between quasiconvexity and k-quasiconvexity, k ≥ 2, is investigated. It is shown that every smooth strictly k-quasiconvex integrand with p-growth at infinity, p > 1, is the restriction to k-th order symmetric tensors of a quasiconvex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semi...

Quasiconvexity of subsurface groups in hyperbolic 3-manifolds

Let Σ be a closed orientable surface of genus g ≥ 2. Suppose that Γ = π1(Σ) acts properly discontinuously on H by orientation preserving isometries. The quotient, M = H/Γ is a complete hyperbolic 3-manifold with a natural homotopy equivalence to Σ. In fact, by tameness [Bon], we know that M is homeomorphic to Σ×R. The proof of the Ending Lamination Conjecture [Mi1,BrCM] has lead to a reasonably...

A Non-quasiconvexity Embedding Theorem for Hyperbolic Groups

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G∗, such that G is a subgroup of G∗ but G is not quasiconvex in G∗ . We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank two which is malnormal and quasiconvex in G.

Higher Order Quasiconvexity Reduces to Quasiconvexity

In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p > 1, is the restriction to symmetric matrices of a 1-quasiconvex functio...

Admissible Orbits of Exceptional Groups 3