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An infinite sequence of ideal hyperbolic Coxeter 4-polytopes and Perron numbers. (English) Zbl 1448.20033

Summary: In [Eur. J. Comb. 32, No. 8, 1299–1316 (2011; Zbl 1242.20049)], R. Kellerhals and G. Perren conjectured that the growth rates of cocompact hyperbolic Coxeter groups are Perron numbers. By results of J. W. Cannon and P. Wagreich [Math. Ann. 293, No. 2, 239–258 (1992; Zbl 0734.57001)], W. J. Floyd [Math. Ann. 293, No. 3, 475–483 (1992; Zbl 0735.51016)], A. Kolpakov [Eur. J. Comb. 33, No. 8, 1709–1724 (2012; Zbl 1252.51012)], Y. Komori and the author [Proc. Japan Acad., Ser. A 91, No. 10, 155–159 (2015; Zbl 1336.20042)], J. Nonaka and R. Kellerhals [Tokyo J. Math. 40, No. 2, 379–391 (2017; Zbl 06855941)], W. Parry [J. Algebra 154, No. 2, 406–415 (1993; Zbl 0796.20031)], the author [RIMS Kôkyûroku Bessatsu B66, 147–165 (2017; Zbl 1388.20057); Can. Math. Bull. 61, No. 2, 405–422 (2018; Zbl 06894636)], the growth rates of 2- and 3-dimensional hyperbolic Coxeter groups are always Perron numbers. A. Kolpakov and A. Talambutsa showed that the growth rates of right-angled Coxeter groups are Perron numbers [Discrete Math. 343, No. 3, Article ID 111763, 8 p. (2020; Zbl 1485.20100)]. For certain families of 4-dimensional cocompact hyperbolic Coxeter groups, the conjecture holds as well (see [R. Kellerhals and G. Perren, Eur. J. Comb. 32, No. 8, 1299–1316 (2011; Zbl 1242.20049); Y. Umemoto, Algebr. Geom. Topol. 14, No. 5, 2721–2746 (2014; Zbl 1307.20036); T. Zehrt and C. Zehrt-Liebendörfer, Beitr. Algebra Geom. 53, No. 2, 451–460 (2012; Zbl 1261.20039)]). In this paper, we construct an infinite sequence of ideal non-simple hyperbolic Coxeter 4-polytopes giving rise to growth rates which are distinct Perron numbers. This is the first explicit example of an infinite family of non-compact finite volume Coxeter polytopes in hyperbolic 4-space whose growth rates are of the conjectured arithmetic nature as well.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F65 Geometric group theory
51M10 Hyperbolic and elliptic geometries (general) and generalizations
20F69 Asymptotic properties of groups
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