Andrew E. Silverio | Searcharxiv

arXiv2016

Reversing Palindromic Enumeration in Rank Two Free Groups

The Gilman-Maskit algorithm for determining the discreteness or non-discreteness of a two-generator subgroup of $\mathrm{PSL}_2\mathbb{R}$ terminates with a pair of generators that are Farey words. The Farey words are primitive words that are indexed by rational numbers and infinity. The E-words, primitive words with palindromic product forms, are also indexed by rational numbers and infinity. We show a modification of the Gilman-Maskit algorithm so that the stopping generators are E-words.

math.GR↗

arXiv2015

Half-turn linked pairs of isometries of hyperbolic 4-space

In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop a new theory of hyperbolic pencils and twisting planes involving a new geometric construction, their half-turn banks. This enables us to give complete results about each of the pair-types of isometries and their simultaneous factorization by half-turns. That is, we provide geometric conditions for each such pair to be linked by half-turns. The main result gives a necessary and sufficient condition for any given pair of isometries to be linked. We also provide a procedure for constructing a half-turn linked pair of isometries of $\mathbb{H}^4$ that do not restrict to lower dimensions, yielding an example that gives a negative answer to a question raised by Ara Basmajian and Karan Puri.

math.MG↗