ABSTRACT. Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic 3-manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass. Pietrowski and Soli tar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.
ABSTRACT. Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic 3-manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass. Pietrowski and Soli tar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.
Imposing restrictions on the allowed sequences of transformations of a standard IFS can give rise to attractors that are attractors of standard IFS consisting of a (larger) finite set of transformations, or a countably infinite set of transformations, or are not the attractor of any standard IFS. Whichever the case it can be read off from the transition graph of the restrictions. * 257 Fractals 1999.07:257-266. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only. √ 3) log(1/2) . The comparison of these calculations was one of the motivations for our explorations. The Rellick, Edgar, Klapper calculation, incorporating which T i can follow which T j , gives the same result as a Moran equation on a larger Fractals 1999.07:257-266. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only.
We demonstrate that videofeedback augmented with one or two mirrors can produce stationary fractal patterns obtained by an iterated function system (IFS) with two transformations (one with a reflection, one without), and with four transformations (two with reflections, one without, one with a 180° rotation). Camera placement yielding only a partial image in one or both mirrors can be achieved using IFS with memory. The IFS rules are obtained from the images of three non-collinear points in each of the principal pieces of the image.
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