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Adian 90: Conference on Mathematical Logic, Algebra, and Computation
July 5, 2021 16:15–17:00, Moscow, Steklov Mathematical Institute of RAS (Moscow) and online in Zoom

Growth rates of Coxeter groups and Perron numbers

A. L. Talambutsa

Steklov Mathematical Institute, Moscow
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Abstract: A classical formula obtained by Steinberg in 1968 shows that the growth series of a Coxeter group (with respect to its standard generating set consisting of involutions) is a rational function, hence the growth rates of these groups are algebraic numbers. In 1980’s Cannon discovered a remarkable connection between Salem polynomials and growth functions of surface groups and some cocompact Coxeter groups. Later Floyd and Parry obtained remarkable results that in many cases the growth rates of cocompact and cofinite Coxeter groups are either Salem or Pisot numbers, which belong to a wider class of Perron numbers. According to the conjecture of Kellerhals–Perren, all standard growth rates of all cofinite Coxeter groups are Perron numbers. For many classes of such groups the conjecture has been confirmed in two recent decades by different authors, however all these results were obtained using the recursive Steinberg's formula, so the groups with big number of generators (more than 15) could not be treated.
We define a large class of abstract Coxeter groups, that we call $\infty$–spanned, and for which we prove that the word growth rate and the geodesic growth rate are Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, including a notable example of Vinberg and Kaplinskaya of a cofinite 50-generated group which acts on the 19-dimensional hyperbolic space. In order to overcome the computational difficulties, instead of Steinberg formula, we use finite automata constructed by Brink and Howlett and Perron-Frobenius theory.
Joint work with Alexander Kolpakov.

Language: English

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