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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.

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
Full Text: DOI arXiv
[1] Bartholdi, L., A Wilson group of non-uniformly exponential growth, Comptes Rendus Mathématique. Académie des Sciences. Paris, 336, 549-554, (2003) · 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 0236.20032
[4] Bucher, M.; de la Harpe, P., Free products with amalgamation and HNN-extensions of uniformly exponential growth, Mathematical Notes, 67, 686-689, (2000) · 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] Eskin, A.; Mozes, S.; Oh, H., On uniform exponential growth for linear groups, Inventiones Mathematicae, 160, 1-30, (2005) · Zbl 1137.20024
[9] Floyd, W. J., Growth of planar Coxeter groups, P. V. numbers, and Salem numbers, Mathematische Annalen, 293, 475-483, (1992) · Zbl 0735.51016
[10] Grigorchuk, R. I.; Harpe, P., One-relator groups of exponential growth have uniformly exponential growth, Mathematical Notes, 69, 628-630, (2001) · Zbl 0999.20033
[11] Harpe, P., Topics in geometric group theory, (2000), Chicago, IL · Zbl 0965.20025
[12] Harpe, P., Uniform growth in groups of exponential growth, Geometriae Dedicata, 95, 1-17, (2002) · Zbl 1025.20027
[13] Gaboriau, D., Invariants l\^{2} de relations d’équivalence et de groupes, Publications Mathématiques. Institut de Hautes Études Scientifiques, 95, 93-150, (2002) · Zbl 1022.37002
[14] Hironaka, E., The Lehmer polynomial and pretzel links, Canadian Mathematical Bulletin, 44, 440-451, (2001) · Zbl 0999.57001
[15] Kellerhals, R., Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers, Algebraic and Geometric Topology, 13, 1001-1025, (2013) · Zbl 1281.20044
[16] Koubi, M., Croissance uniforme dans LES groupes hyperboliques, Université de Grenoble. Annales de l’Institut Fourier, 48, 1441-1453, (1998) · Zbl 0914.20033
[17] Lyons, R.; Pichot, M.; Vassout, S., Uniform non-amenability, cost, and the first l2-Betti number, Geometry, Groups, and Dynamics, 2, 595-617, (2008) · Zbl 1196.37011
[18] Mann, A., The growth of free products, Journal of Algebra, 326, 208-217, (2011) · 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] Nekrashevych, V., A group of non-uniform exponential growth locally isomorphic to IMG(z\^{2} + i), Transactions of the American Mathematical Society, 362, 389-398, (2010) · Zbl 1275.20049
[21] Osin, D., The entropy of solvable groups, Ergodic Theory and Dynamical Systems, 23, 907-918, (2003) · Zbl 1062.20039
[22] J.-P. Serre, Trees, Springer-Verlag, Berlin-New York, 1980. · Zbl 0548.20018
[23] Siegel, C., Algebraic numbers whose conjugates Lie in the unit circle, Duke Mathemaatical Journal, 11, 597-602, (1944) · Zbl 0063.07005
[24] Talambutsa, A., Attainability of the minimal exponential growth rate for free products of finite cyclic groups, Trudy Mtematicheskogo Instituta Imeni V.A. Steklova, 274, 289-302, (2011) · Zbl 1297.20024
[25] Tits, J., Free subgroups in linear groups, Journal of Algebra, 20, 250-270, (1972) · Zbl 0236.20032
[26] Wilson, J. S., On exponential growth and uniformly exponential growth for groups, Inventiones Mathematicae, 155, 287-303, (2004) · Zbl 1065.20054
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