Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Kellerhals, Ruth

doi:10.2140/agt.2013.13.1001

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Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Ruth Kellerhals

Algebraic & Geometric Topology 13 (2013) 1001–1025

DOI: 10.2140/agt.2013.13.1001

Abstract

By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of $H^{3}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>3</mn></mrow></msup></math>$ by the tetrahedral Coxeter group $(3, 3, 6)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mn>3</mn><mo class="MathClass-punc">,</mo><mn>3</mn><mo class="MathClass-punc">,</mo><mn>6</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ has minimal volume. We prove that the group $(3, 3, 6)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mn>3</mn><mo class="MathClass-punc">,</mo><mn>3</mn><mo class="MathClass-punc">,</mo><mn>6</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ 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 $(3, \infty)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mn>3</mn><mo class="MathClass-punc">,</mo><mi>∞</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ 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 $(3, 3, 6)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mn>3</mn><mo class="MathClass-punc">,</mo><mn>3</mn><mo class="MathClass-punc">,</mo><mn>6</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is not a Pisot number.

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)