×

On the growth of hyperbolic 3-dimensional generalized simplex reflection groups. (English) Zbl 1244.20039

Summary: We prove that the growth rates of three-dimensional generalized simplex reflection groups, i.e. three-dimensional non-compact hyperbolic Coxeter groups with four generators are always Perron numbers.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20F67 Hyperbolic groups and nonpositively curved groups
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] J. W. Cannon and Ph. Wagreich, Growth functions of surface groups, Math. Ann. 293 (1992), no. 2, 239-257. · Zbl 0734.57001 · doi:10.1007/BF01444714
[2] W. J. Floyd, Growth of planar Coxeter groups, P.V. numbers, and Salem numbers, Math. Ann. 293 (1992), no. 3, 475-483. · Zbl 0735.51016 · doi:10.1007/BF01444729
[3] P. de la Harpe, Groupes de Coxeter infinis non affines, Exposition. Math. 5 (1987), no. 1, 91-96. · Zbl 0605.20049
[4] G. J. Heckman, The volume of hyperbolic Coxeter polytopes of even dimension, Indag. Math. (N.S.) 6 (1995), no. 2, 189-196. · Zbl 0831.51007 · doi:10.1016/0019-3577(95)91242-N
[5] J. E. Humphreys, Reflection groups and Coxeter groups , Cambridge Studies in Advanced Mathematics, 29, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0725.20028
[6] R. Kellerhals and G. Perren, On the growth of cocompact hyperbolic Coxeter groups, European J. Combin. 32 (2011), no. 8, 1299-1316. · Zbl 1242.20049 · doi:10.1016/j.ejc.2011.03.020
[7] Y. Komori and Y. Umemoto, On the growth of 3-dimensional hyperbolic Coxeter groups with 4 and 5 generators. (in preparation). · Zbl 1244.20039
[8] W. Parry, Growth series of Coxeter groups and Salem numbers, J. Algebra 154 (1993), no. 2, 406-415. · Zbl 0796.20031 · doi:10.1006/jabr.1993.1022
[9] J. G. Ratcliffe, Foundations of hyperbolic manifolds , Grad. Texts in Math., 149, Springer, New York, 1994. · Zbl 0809.51001
[10] J.-P. Serre, Cohomologie des groupes discrets, in Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, NJ, 1970) , 77-169. Ann. of Math. Studies, 70, Princeton Univ. Press, Princeton, NJ, 1971. · Zbl 0235.22020
[11] L. Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376-393. · Zbl 0151.02003 · doi:10.1016/0021-8693(66)90007-X
[12] R. Steinberg, Endomorphisms of linear algebraic groups , Memoirs of the American Mathematical Society, No. 80, Amer. Math. Soc., Providence, RI, 1968. · Zbl 0164.02902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.