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- A. HINKKANEN, G. J. MARTIN
- 1996

This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces… (More)

@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)

- PETRA BONFERT-TAYLOR, RICHARD D. CANARY, GAVEN MARTIN, EDWARD TAYLOR
- 2004

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)

- F. W. Gehring, C. Maclachlan, G. J. Martin, A. W. Reid
- 1997

We give an arithmetic criterion which is sufficient to imply the dis-creteness of various two-generator subgroups of P SL(2, C). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis… (More)

- PETRA BONFERT-TAYLOR, RICHARD D. CANARY, GAVEN MARTIN, EDWARD C. TAYLOR, MICHAEL WOLF
- 2009

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)

- GAVEN J. MARTIN, BRAD G. OSGOOD, F. W. Gehring
- 2007

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)

- G. J. MARTIN
- 1997

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)

- Frederick W. Gehring, Petra Bonfert-Taylor, Gaven Martin, Alan W. Reid, Edward C. Taylor, P. Bonfert-Taylor +3 others
- 2010

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)

- M R Bridson, D Groves, J A Hillman, G J Martin
- 2009

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)