Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of $ℍ^{3}$ by the tetrahedral Coxeter group $(3, 3, 6)$ has minimal volume. We prove that the group $(3, 3, 6)$ has smallest growth rate among all non-cocompact cofinite hyperbolic Coxeter groups, and that it is as such unique. This result extends to three dimensions some work of W Floyd who showed that the Coxeter triangle group $(3, \infty)$ has minimal growth rate among all non-cocompact cofinite planar hyperbolic Coxeter groups. In contrast to Floyd’s result, the growth rate of the tetrahedral group $(3, 3, 6)$ is not a Pisot number.

PDF Access Denied

Warning: We have not been able to recognize your IP address 47.88.87.18 as that of a subscriber to this journal. Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form.

Or, visit our subscription page for instructions on purchasing a subscription. You may also contact us at contact@msp.org or by using our contact form.

Or, you may purchase this single article for USD 29.95:

Keywords

Hyperbolic Coxeter group, cusp, growth rates, Pisot number

Mathematical Subject Classification 2000

Primary: 20F55

Secondary: 22E40, 51F15

References

Publication

Received: 12 July 2012

Revised: 30 November 2012

Accepted: 5 December 2012

Published: 5 April 2013

Authors

Ruth Kellerhals
Department of Mathematics University of Fribourg CH-1700 Fribourg Switzerland

Volume 13, issue 2 (2013)

Ruth Kellerhals

Abstract