Modular circle quotients and PL limit sets

We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal boundary of H^2. Given two points p,q in S^1 we write pq if p and q are the endpoints of a geodesic in Gamma. (In particular pp.) We show that is an equivalence relation. We let Q_Gamma=S^1/ be the quotient space. We call Q_Gamma a modular circle quotient. In this paper we will give a sense of what modular circle quotients `look like' by realizing them as limit sets of piecewise-linear group actionsComment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper1.abs.htm

A Compiler Algorithm for Managing Asynchronous Memory Read Completion

Computers with conventional memory systems have a predictable latency between initiation and completion of a memory read. On such machines it is relatively easy for either the compiler or the processor to guarantee that a load has completed before further references to the loaded register are made. In a machine with a logically shared, but physically distributed, memory, these latencies are not statically predictable. Some existing systems, such as the Cray T3D, deal with this problem by using a hardware mechanism to enforce synchronization on a register which is the target of a remote memory access. The M-Machine currently being designed by the Concurrent VLSI Architecture Group at MIT performs remote memory accesses asynchronously, and allows program execution to continue while the access is outstanding, but does not enforce synchronization in hardware. This architectural simplification, and resulting relaxation of memory completion semantics, poses a challenge to the compiler: how can this simpler memory system be efficiently supported while maintaining program correctness. In particular, what is required to guarantee that there are no conflicts between completion of a memory operation by placing a value into a register, and other uses of the register being written. This paper describes a general solution to this problem, develops an algorithm to implement it, and shows that the algorithm is correct

Teichmuller geodesics that do not have a limit in PMF

We construct a Teichmuller geodesic which does not have a limit on the Thurston boundary of the Teichmuller space.Comment: published versio

A Message-Driven Programming System for Fine-Grain Multicomputers

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Algebraic invariants, mutation, and commensurability of link complements

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some have integral traces while others do not.Comment: Minor changes following referee's suggestion

Recurrence and pressure for group extensions

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterises recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincar\'e series of Kleinian groups and the cogrowth of group presentations

Projective structures, grafting, and measured laminations

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective (\CP^1) structures on a surface. We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.Comment: 31 pages, 4 figure

Convex cocompact subgroups of mapping class groups

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a "Schottky subgroup" of MCG, the converse is true as well; a semidirect product of pi_1(S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G=Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper5.abs.htm