Universal Reflection Subgroups and Exponential Growth in Coxeter Groups

We investigate the imaginary cone in hyperbolic Coxeter systems in order to show that any Coxeter system contains universal reflection subgroups of arbitrarily large rank. Furthermore, in the hyperbolic case, the positive spans of the simple roots of the universal reflection subgroups are shown to approximate the imaginary cone (using an appropriate topology on the set of roots), answering a question due to Dyer 9 Dyer , M. Imaginary Cone and Reflection Subgroups of Coxeter Groups. Preprint: http://arXiv.org/abs/1210.5206  Google Scholar]] in the special case of hyperbolic Coxeter systems. Finally, we discuss growth in Coxeter systems and utilize the previous results to extend the results of 16 Viswanath , S. ( 2008 ). On growth types of quotients of Coxeter groups by parabolic subgroups . Comm. Algebra 36 ( 2 ): 796 – 805 .Taylor &; Francis Online] , Google Scholar]] regarding exponential growth in parabolic quotients in Coxeter groups.