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Search: All articles in the CJM digital archive with keyword Salem number

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1. CJM 2013 (vol 66 pp. 354)

Kellerhals, Ruth; Kolpakov, Alexander
The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space
Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is as such unique. Our approach provides a different proof for the analog situation in ${\mathbb H^2}$ where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

Keywords:hyperbolic Coxeter group, growth rate, Salem number
Categories:20F55, 22E40, 51F15

2. CJM 2011 (vol 64 pp. 345)

McKee, James; Smyth, Chris
Salem Numbers and Pisot Numbers via Interlacing
We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the ``obvious'' limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

Keywords:Salem numbers, Pisot numbers
Category:11R06

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