Heegaard genus and property τ for hyperbolic 3‐manifolds

A display package for holding a folded box for audio-visual material having indicia describing the box contents which consists of a transparent envelope, the envelope comprising sheets of plastic material secured together on their edges, one edge being open to permit the folded box to be inserted into and withdrawn from the envelope, one sheet being rigid for holding the envelope upstanding, thereby being a stop seam at one end of the envelope to hold the folded box positioned away from the envelope edge, and a pocket secured to the envelope on its end opposite the stop seam for holding information related to the category of the audio-visual material. The invention also relates to an indexing system for audio-visual material where a plurality of like such display packages are arranged one behind another in a bin, packed loosely to allow one package to be flipped away from another package for selective browsing through the packages, and to a method for arranging such a system which consists of the steps of removing the materials from their boxes, folding boxes to a flattened condition, inserting each box into such a display package and arranging the packages uprights in a bin for browsing therethrough by a consumer. The display package pocket also includes novel means for holding indicia placed therein from falling from the pocket during flipping and handling of the packages.

[1]  R. Brooks Circle packings and co-compact extensions of Kleinian groups , 1986 .

[2]  G. A. Margulis,et al.  Explicit constructions of graphs without short cycles and low density codes , 1982, Comb..

[3]  H. Helfgott Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$ , 2008 .

[4]  Terence Tao,et al.  A sum-product estimate in finite fields, and applications , 2003, math/0301343.

[5]  L. Clozel Démonstration de la conjecture τ , 2003 .

[6]  A. Reid,et al.  Simple quotients of hyperbolic 3-manifold groups , 1998 .

[7]  A. Lubotzky,et al.  On groups of polynomial subgroup growth , 1991 .

[8]  A. Reid,et al.  The Arithmetic of Hyperbolic 3-Manifolds , 2002 .

[9]  Marc Lackenby Heegaard splittings, the virtually Haken conjecture and Property (τ) , 2002 .

[10]  EIGENVALUES OF THE LAPLACIAN, THE FIRST BETTI NUMBER AND THE CONGRUENCE SUBGROUP PROBLEM , 1996 .

[11]  A. Reid,et al.  Rigidity of cusps in deformations of hyperbolic 3-orbifolds , 1993 .

[12]  James W. Anderson A brief survey of the deformation theory of Kleinian groups , 1998, math/9810186.

[13]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[14]  Jean Bourgain,et al.  Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order , 2006 .

[15]  P. Sarnak,et al.  Sieving and expanders , 2006 .

[16]  M. Gromov,et al.  Non-arithmetic groups in lobachevsky spaces , 1987 .

[17]  P. Sarnak,et al.  Ramanujan duals II , 1991 .

[18]  Patrick D. Shanahan,et al.  TRACE FIELDS OF TWIST KNOTS , 2001 .

[19]  J. Bourgain,et al.  Uniform expansion bounds for Cayley graphs of SL2(Fp) , 2008 .