×

Exponential growth rates of free and amalgamated products. (English) Zbl 1350.20020

Summary: We prove that there is a gap between \(\sqrt 2\) and \((1+\sqrt 5)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products \(G=A*_CB\) with \(([A:C]-1)([B:C]-1)\geq 2\), we show that an exponential growth rate lower than \(\sqrt 2 \) can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to \(\psi\approx 1.325\), where \(\psi\) is the unique positive root of the polynomial \(z^3-z-1\). One of these groups is \(\mathrm{PGL}(2,\mathbb Z)\cong(C_2\times C_2)*_{C_2}D_6\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2\). This answers two questions by A. Mann [J. Algebra 326, No. 1, 208-217 (2011; Zbl 1231.20027)]. We also prove that \(\psi\) is a lower bound for the minimal growth rates of a large class of Coxeter groups, including cofinite non-cocompact planar hyperbolic groups, which strengthens a result obtained earlier by William Floyd, who considered only standard Coxeter generators.

MSC:

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05 Generators, relations, and presentations of groups
20F69 Asymptotic properties of groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L. Bartholdi, A Wilson group of non-uniformly exponential growth, Comptes Rendus Mathématique. Académie des Sciences. Paris 336 (2003), 549-554. · Zbl 1050.20018
[2] L. Bieberbach, Analytische Fortsetzung. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 3. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. · Zbl 0064.06902
[3] A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005. · Zbl 1110.05001
[4] M. Bucher and P. de la Harpe, Free products with amalgamation and HNN-extensions of uniformly exponential growth, Mathematical Notes 67 (2000), 686-689, translated from Mateaticheskie Zametki 67 (2000), 811-815. · Zbl 0998.20025
[5] Chiswell, I., Introduction to Λ-trees (2001) · Zbl 1004.20014
[6] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 3rd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 14, Springer-Verlag, New York-Heidelberg, 1972. · Zbl 0239.20040
[7] W. Dicks and M. J. Dunwoody, Groups Acting on Graphs, Cambridge Studies in Advanced Mathemaics, Vol. 17, Cambridge University Press, Cambridge, 1989. · Zbl 0665.20001
[8] A. Eskin, S. Mozes and H. Oh, On uniform exponential growth for linear groups, Inventiones Mathematicae 160 (2005), 1-30. · Zbl 1137.20024
[9] W. J. Floyd, Growth of planar Coxeter groups, P. V. numbers, and Salem numbers, Mathematische Annalen 293 (1992), 475-483. · Zbl 0735.51016
[10] R. I. Grigorchuk and P. de la Harpe, One-relator groups of exponential growth have uniformly exponential growth, Mathematical Notes 69 (2001), 628-630. · Zbl 0999.20033
[11] Harpe, P., Topics in Geometric Group Theory (2000), Chicago, IL · Zbl 0965.20025
[12] P. de la Harpe, Uniform growth in groups of exponential growth, Geometriae Dedicata 95 (2002), 1-17. · Zbl 1025.20027
[13] D. Gaboriau, Invariants l2de relations d’équivalence et de groupes, Publications Mathématiques. Institut de Hautes Études Scientifiques 95 (2002), 93-150. · Zbl 1022.37002
[14] E. Hironaka, The Lehmer polynomial and pretzel links, Canadian Mathematical Bulletin 44 (2001), 440-451. · Zbl 0999.57001
[15] R. Kellerhals, Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers, Algebraic and Geometric Topology 13 (2013), 1001-1025. · Zbl 1281.20044
[16] M. Koubi, Croissance uniforme dans les groupes hyperboliques, Université de Grenoble. Annales de l’Institut Fourier 48 (1998), 1441-1453. · Zbl 0914.20033
[17] R. Lyons, M. Pichot and S. Vassout, Uniform non-amenability, cost, and the first l2-Betti number, Geometry, Groups, and Dynamics 2 (2008), 595-617. · Zbl 1196.37011
[18] A. Mann, The growth of free products, Journal of Algebra 326 (2011), 208-217. · Zbl 1231.20027
[19] A. Mann, How Groups Grow, London Mathematical Society Lecture Note Series, Vol. 395, Cambridge University Press, Cambridge, 2012. · Zbl 1253.20032
[20] V. Nekrashevych, A group of non-uniform exponential growth locally isomorphic to IMG(z2 + i), Transactions of the American Mathematical Society 362 (2010), 389-398. · Zbl 1275.20049
[21] D. Osin, The entropy of solvable groups, Ergodic Theory and Dynamical Systems 23 (2003), 907-918. · Zbl 1062.20039
[22] J.-P. Serre, Trees, Springer-Verlag, Berlin-New York, 1980. · Zbl 0548.20018
[23] C. Siegel, Algebraic numbers whose conjugates lie in the unit circle, Duke Mathemaatical Journal 11 (1944), 597-602. · Zbl 0063.07005
[24] A. Talambutsa, Attainability of the minimal exponential growth rate for free products of finite cyclic groups, Trudy Mtematicheskogo Instituta Imeni V.A. Steklova 274 (2011), 289-302. · Zbl 1297.20024
[25] J. Tits, Free subgroups in linear groups, Journal of Algebra 20 (1972), 250-270. · Zbl 0236.20032
[26] J. S. Wilson, On exponential growth and uniformly exponential growth for groups, Inventiones Mathematicae 155 (2004), 287-303. · Zbl 1065.20054
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.