Gaven J. Martin

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In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)-'ldx I. Due to the fact that the density cl(x, aD)-t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic(More)
Let n ≥ 2 and f : S n → S n be a quasiregular mapping with branch set B f , the set where f fails to be locally injective. We show that there is a quasiregular mapping g : S n → S n with Bg = B f and such that g can be chosen to be conformal (rational) with respect to some measurable Riemannian structure on S n. Hence g is uniformly quasiregular. That is, g(More)
We show that there exists a universal constant K c so that every K-strongly quasiconformally homogeneous hyperbolic surface X (not equal to H 2) has the property that K > K c > 1. The constant K c is the best possible, and is computed in terms of the diameter of the (2, 3, 7)-hyperbolic orbifold (which is the hyperbolic orbifold of smallest area.) We(More)
A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an auto-morphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is(More)
The Margulis constant for Kleinian groups is the smallest constant c such that for each discrete group G and each point x in the upper half space H 3 , the group generated by the elements in G which move x less than distance c is elementary. We take a first step towards determining this constant by proving that if f, g is nonelementary and discrete with f(More)
Editor's note: Gaven Martin kindly agreed to write for us an introduction to the story he presented in his invited lecture " Siegel's Problem on Small Volume Lattices " at the November AMS Fall Southeastern Sectional Meeting held at the University of St. Thomas in Minneapolis. Abstract. We discuss our recent solution to Siegel's 1943 problem concerning the(More)
The existence and uniqueness properties for extremal mappings with smallest weighted L p distortion between annuli and the related Grötzsch type problems are discussed. An interesting critical phase type phenomena is observed. When p < 1, apart from the identity map, minimizers never exist. When p = 1 we observe Nitsche type phenomena; minimisers exist(More)