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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.

##### 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
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##### References:
  Bartholdi, L., A Wilson group of non-uniformly exponential growth, Comptes Rendus Mathématique. Académie des Sciences. Paris, 336, 549-554, (2003) · Zbl 1050.20018  L. Bieberbach, Analytische Fortsetzung. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 3. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. · Zbl 0064.06902  A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005. · Zbl 0236.20032  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  Chiswell, I., Introduction to λ-trees, (2001) · Zbl 1004.20014  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  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  Eskin, A.; Mozes, S.; Oh, H., On uniform exponential growth for linear groups, Inventiones Mathematicae, 160, 1-30, (2005) · Zbl 1137.20024  Floyd, W. J., Growth of planar Coxeter groups, P. V. numbers, and Salem numbers, Mathematische Annalen, 293, 475-483, (1992) · Zbl 0735.51016  Grigorchuk, R. I.; Harpe, P., One-relator groups of exponential growth have uniformly exponential growth, Mathematical Notes, 69, 628-630, (2001) · Zbl 0999.20033  Harpe, P., Topics in geometric group theory, (2000), Chicago, IL · Zbl 0965.20025  Harpe, P., Uniform growth in groups of exponential growth, Geometriae Dedicata, 95, 1-17, (2002) · Zbl 1025.20027  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  Hironaka, E., The Lehmer polynomial and pretzel links, Canadian Mathematical Bulletin, 44, 440-451, (2001) · Zbl 0999.57001  Kellerhals, R., Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers, Algebraic and Geometric Topology, 13, 1001-1025, (2013) · Zbl 1281.20044  Koubi, M., Croissance uniforme dans LES groupes hyperboliques, Université de Grenoble. Annales de l’Institut Fourier, 48, 1441-1453, (1998) · Zbl 0914.20033  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  Mann, A., The growth of free products, Journal of Algebra, 326, 208-217, (2011) · Zbl 1231.20027  A. Mann, How Groups Grow, London Mathematical Society Lecture Note Series, Vol. 395, Cambridge University Press, Cambridge, 2012. · Zbl 1253.20032  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  Osin, D., The entropy of solvable groups, Ergodic Theory and Dynamical Systems, 23, 907-918, (2003) · Zbl 1062.20039  J.-P. Serre, Trees, Springer-Verlag, Berlin-New York, 1980. · Zbl 0548.20018  Siegel, C., Algebraic numbers whose conjugates Lie in the unit circle, Duke Mathemaatical Journal, 11, 597-602, (1944) · Zbl 0063.07005  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  Tits, J., Free subgroups in linear groups, Journal of Algebra, 20, 250-270, (1972) · Zbl 0236.20032  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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