Algebraic & Geometric Topology

The growth function of Coxeter dominoes and $2$–Salem numbers

Yuriko Umemoto

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By the results of Cannon, Wagreich and Parry, it is known that the growth rate of a cocompact Coxeter group in 2 and 3 is a Salem number. Kerada defined a j–Salem number, which is a generalization of Salem numbers. In this paper, we realize infinitely many 2–Salem numbers as the growth rates of cocompact Coxeter groups in 4. Our Coxeter polytopes are constructed by successive gluing of Coxeter polytopes, which we call Coxeter dominoes.

Article information

Algebr. Geom. Topol., Volume 14, Number 5 (2014), 2721-2746.

Received: 20 May 2013
Revised: 4 September 2013
Accepted: 10 September 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]

hyperbolic Coxeter group growth rate $2$–Salem number


Umemoto, Yuriko. The growth function of Coxeter dominoes and $2$–Salem numbers. Algebr. Geom. Topol. 14 (2014), no. 5, 2721--2746. doi:10.2140/agt.2014.14.2721.

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  • M-J Bertin, A Decomps-Guilloux, M Grandet-Hugot, M Pathiaux-Delefosse, J-P Schreiber, Pisot and Salem numbers, Birkhäuser, Basel (1992)
  • J W Cannon, P Wagreich, Growth functions of surface groups, Math. Ann. 293 (1992) 239–257
  • R Charney, M Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991) 373–378
  • H S M Coxeter, Discrete groups generated by reflections, Ann. of Math. 35 (1934) 588–621
  • E Ghate, E Hironaka, The arithmetic and geometry of Salem numbers, Bull. Amer. Math. Soc. 38 (2001) 293–314
  • J E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies Adv. Math. 29, Cambridge Univ. Press (1990)
  • R Kellerhals, G Perren, On the growth of cocompact hyperbolic Coxeter groups, European J. Combin. 32 (2011) 1299–1316
  • A J Kempner, On the complex roots of algebraic equations, Bull. Amer. Math. Soc. 41 (1935) 809–843
  • M Kerada, Une caractérisation de certaines classes d'entiers algébriques généralisant les nombres de Salem, Acta Arith. 72 (1995) 55–65
  • F Lannér, On complexes with transitive groups of automorphisms, Comm. Sém., Math. Univ. Lund 11 (1950) 71
  • D H Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. 34 (1933) 461–479
  • F Luo, On a problem of Fenchel, Geom. Dedicata 64 (1997) 277–282
  • V S Makarov, The Fedorov groups of four-dimensional and five-dimensional Lobačevskiĭ space, from: “Studies in general algebra”, (V D Belousov, editor), Kišinev. Gos. Univ., Kishinev (1968) 120–129
  • M R Murty, Prime numbers and irreducible polynomials, Amer. Math. Monthly 109 (2002) 452–458
  • W Parry, Growth series of Coxeter groups and Salem numbers, J. Algebra 154 (1993) 406–415
  • J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer, New York (1994)
  • P A Samet, Algebraic integers with two conjugates outside the unit circle, Proc. Cambridge Philos. Soc. 49 (1953) 421–436
  • I Satake, Linear algebra, Pure and Applied Mathematics 29, Marcel Dekker, New York (1975)
  • L Schlettwein, Hyperbolische simplexe, Diplomarbeit, Universität Basel (1995)
  • J-P Serre, Cohomologie des groupes discrets, from: “Prospects in mathematics”, Ann. of Math. Studies 70, Princeton Univ. Press (1971) 77–169
  • L Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966) 376–393
  • R Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the AMS 80, Amer. Math. Soc. (1968)
  • È B Vinberg, Hyperbolic groups of reflections, Uspekhi Mat. Nauk 40 (1985) 29–66, 255 In Russian; translated in Russian Math. Surveys 40 (1985) 31–75
  • R L Worthington, The growth series of compact hyperbolic Coxeter groups with $4$ and $5$ generators, Canad. Math. Bull. 41 (1998) 231–239
  • T Zehrt, C Zehrt-Liebend örfer, The growth function of Coxeter garlands in $\mathbb{H}\sp 4$, Beitr. Algebra Geom. 53 (2012) 451–460