Gaven J. Martin

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@0-categorical o-minimal structures were completely described in 4], and are essentially built up from copies of the rationals as an ordered set by`cutting and copying'. Here we investigate the possible structures which an @0-categorical weakly o-minimal set may carry, and nd that there are some rather more interesting (and not o-minimal) examples. We show(More)
We exhibit strong constraints on the geometry and topology of a uniformly quasiconformally homogeneous hyperbolic manifold. In particular, if n ≥ 3, a hyperbolic n-manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. Moreover, if n ≥ 3, we show that there is a constant K n > 1 such that if(More)
Every finite group acts as the full isometry group of some compact hyperbolic 3-manifold. In this paper we study those finite groups which act maximally, that is when the ratio |Isom + (M)|/vol(M) is maximal among all such manifolds. In two dimensions maximal symmetry groups are called Hurwitz groups, and arise as quotients of the (2,3,7)–triangle group.(More)
We prove that the ambient quasiconformal homogene-ity constant of a hyperbolic planar domain which is not simply connected is uniformly bounded away from 1. We also consider a component Ω 0 of a finitely generated Kleinian group Γ. We show that if Ω 0 /Γ is compact, then Ω 0 is uniformly ambiently quasiconformally homogeneous, and that if Ω 0 is not simply(More)
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)