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Cogrowth of groups with small cancellation. (Cocroissance des groupes à petite simplification.) (French) Zbl 0829.20046

Summary: Let \(\Gamma\) be a group presented by \(\langle e_1,\dots,e_m\mid r_1,\dots,r_k\rangle\), \(L\) the free group generated by \(e_1,\dots,e_m\), and \(N=\text{Ker}(L\to\Gamma)\). Let \(c_n\) be the number of elements of length \(n\) in \(N\). We know that \(c=\lim\sup(c_n)^{1/n}\) exists and that \(\sqrt{2m-1}<c\leq 2m-1\) if \(N\neq\{1\}\). We prove that if the group \(\Gamma\) satisfies a condition slightly weaker than the small cancellation condition \(C'(\lambda)\) with \(\lambda<1/6\), then \(c\to\sqrt{2m-1}\) when the lengths of the relations \(r_i\) tend to infinity. A consequence of this result is a theorem of Grigorchuk.

MSC:

20F05 Generators, relations, and presentations of groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F65 Geometric group theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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