Algebraic & Geometric Topology

Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Ruth Kellerhals

Full-text: Open access


By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of 3 by the tetrahedral Coxeter group (3,3,6) has minimal volume. We prove that the group (3,3,6) 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,) 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) is not a Pisot number.

Article information

Algebr. Geom. Topol., Volume 13, Number 2 (2013), 1001-1025.

Received: 12 July 2012
Revised: 30 November 2012
Accepted: 5 December 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]

Hyperbolic Coxeter group cusp growth rates Pisot number


Kellerhals, Ruth. Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers. Algebr. Geom. Topol. 13 (2013), no. 2, 1001--1025. doi:10.2140/agt.2013.13.1001.

Export citation


  • 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
  • H S M Coxeter, W O J Moser, Generators and relations for discrete groups, fourth edition, Ergeb. Math. Grenzgeb. 14, Springer, Berlin (1980)
  • W J Floyd, Growth of planar Coxeter groups, P V numbers, and Salem numbers, Math. Ann. 293 (1992) 475–483
  • P de la Harpe, Groupes de Coxeter infinis non affines, Exposition. Math. 5 (1987) 91–96
  • E Hironaka, The Lehmer polynomial and pretzel links, Canad. Math. Bull. 44 (2001) 440–451
  • N W Johnson, R Kellerhals, J G Ratcliffe, S T Tschantz, The size of a hyperbolic Coxeter simplex, Transform. Groups 4 (1999) 329–353
  • I M Kaplinskaja, The discrete groups that are generated by reflections in the faces of simplicial prisms in Lobačevskiĭ spaces, Mat. Zametki 15 (1974) 159–164
  • R Kellerhals, A Kolpakov, The minimal growth rate of cocompact Coxeter groups in hyperbolic 3–space, to appear in Canadian J. Math.
  • R Kellerhals, G Perren, On the growth of cocompact hyperbolic Coxeter groups, European J. Combin. 32 (2011) 1299–1316
  • A Kolpakov, Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers, European J. Combin. 33 (2012) 1709–1724
  • Y Komori, Y Umemoto, The growth functions of noncompact 3–dimensional hyperbolic Coxeter groups with 4 and 5 generators, preprint (2012)
  • Y Komori, Y Umemoto, On the growth of hyperbolic 3–dimensional generalized simplex reflection groups, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012) 62–65
  • J-L Koszul, Lectures on hyperbolic Coxeter groups, lecture notes, University of Notre Dame (1967)
  • R Meyerhoff, The cusped hyperbolic 3–orbifold of minimum volume, Bull. Amer. Math. Soc. 13 (1985) 154–156
  • W Parry, Growth series of Coxeter groups and Salem numbers, J. Algebra 154 (1993) 406–415
  • J Rotman, Galois theory, second edition, Universitext, Springer, New York (1998)
  • C J Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971) 169–175
  • 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 American Mathematical Society 80, American Mathematical Society (1968)
  • È B Vinberg, Hyperbolic groups of reflections, Uspekhi Mat. Nauk 40 (1985) 29–66, 255
  • È B Vinberg, O V Shvartsman, Discrete groups of motions of spaces of constant curvature, from: “Geometry, II”, Encyclopaedia Math. Sci. 29, Springer, Berlin (1993) 139–248
  • T Zehrt, C Zehrt-Liebend örfer, The growth function of Coxeter garlands in $\mathbb{H}^4$, Beitr. Algebra Geom. 53 (2012) 451–460