Exponential growth rates of free and amalgamated products
| Autor: | Michelle Bucher, Alexey Talambutsa |
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| Rok vydání: | 2016 |
| Předmět: |
Large class
Discrete mathematics Polynomial (hyperelastic model) General Mathematics 010102 general mathematics Coxeter group Group Theory (math.GR) 01 natural sciences Upper and lower bounds Combinatorics Free product Exponential growth 20F65 20E06 20E08 20F69 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Algebra over a field Mathematics - Group Theory Mathematics |
| Zdroj: | Israel Journal of Mathematics. 212:521-546 |
| ISSN: | 1565-8511 0021-2172 |
| Popis: | We prove that there is a gap between $\sqrt{2}$ and $(1+\sqrt{5})/2$ for the exponential growth rate of free products $G=A*B$ not isomorphic to the infinite dihedral group. For amalgamated products $G=A*_C B$ with $([A:C]-1)([B:C]-1)\geq2$, we show that lower exponential growth rate than $\sqrt{2}$ can be achieved by proving that the exponential growth rate of the amalgamated product $\mathrm{PGL}(2,\mathbb{Z})\cong (C_2\times C_2) *_{C_2} D_6$ is equal to the unique positive root of the polynomial $z^3-z-1$. This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011) 208--217]. 17 pages, 7 figures |
| Databáze: | OpenAIRE |
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