By Chaohao G. (ed.), Berger M., Bryant R.L.
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Solution of the Burnside problem for exponent six. Illinois J. Math. 2 (1958), 764–786. H. Neumann, H. Neumann. Embedding Theorems for Groups, J. London Math. Soc. 34 (1959), 465–479. [OS02] A. Olshanskii, M. Sapir. Non-amenable ﬁnitely presented torsion-by-cyclic ´ groups. Publ. Math. Inst. Hautes Etudes Sci. 96 (2002), 43–169. [Sh99] P. Shumyatsky. On groups with commutators of bounded order, Proc. Amer. Math. Soc. 127 (1999), 2583–2586. 50 Y. de Cornulier and A. Mann [Sh02] P. Shumyatsky. Commutators in residually ﬁnite groups, Monatsh.
But the exact growth function of F remains unknown – it is not even known if the growth function is rational, though Cleary, Elder and Taback  show that there are inﬁnitely many cone types, which may be evidence that the growth of the full language of geodesics is not rational. Here, we use a computational approach to estimate the growth function of F . We use two methods both based upon taking random samples of words via random walks. Both of these methods estimate the number of words in successive n-spheres of F .
300, 320, it gives in the second and third columns the sample size (the number of words that were tested) and the number of words among them that were found to represent the trivial element of F ; thus the quotient of these two quantities is an approximation of p(L). The fourth column contains the Lth root of this proportion. The last column contains the 20th root of the quotient of the proportions obtained for length L and for length L − 20. 9161 Table 1. Cogrowth estimates for F . In order to clarify the last two columns we remark that the sequences L p(L) and 20 p(L)/p(L − 20) have the same limits – for instance if we had p(L) const · aL then we would obtain lim L→∞ L p(L) = lim 20 L→∞ p(L)/p(L − 20) = a The diﬀerence between the two sequences is that the second one converges much more quickly, but it is also more sensitive to statistical errors related to insuﬃcient sample size.
Differential Geometry and Differential Equations by Chaohao G. (ed.), Berger M., Bryant R.L.