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Statistical properties of finitely presented groups. (Propriétés statistiques des groupes de présentation finie.) (French) Zbl 0847.20030
Let \(L_m\) be the free group on \(m\) generators \(e_1,\dots,e_m\) and let (P) be a property of the presentations of the groups obtained by adjoining two relators to \(L_m\). For every pair of nonnegative integers \(n_1\) and \(n_2\), let \(p_{n_1,n_2}\) be the proportion of pairs \(r_1\), \(r_2\) of cyclically reduced words in \(L_m\) of length \(n_1\) and \(n_2\), respectively, such that the presentation \(\langle e_1,\dots,e_m\mid r_1,r_2\rangle\) satisfies property (P). Then property (P) is said to hold asymptotically almost surely if \(p_{n_1,n_2}\) converges to 1 when \(\text{inf}\{n_1,n_2\}\) tends to infinity. The main result of this paper is the following Theorem: The presentations of groups with \(m\) generators and 2 relators are asymptotically almost surely word-hyperbolic groups of cohomological dimension 2, with boundary a Menger curve. The paper contains other (weaker) genericity results which hold for groups with a fixed arbitrary number of relators and the author claims that the theorem stated above is probably true for an arbitrary fixed number of relators. The proofs given in this paper rely on ideas and techniques that are sketched in M. Gromov’s paper [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], which contains statements of results of the same sort. Results in the same direction have also been proved independently by A. Yu. Ol’shanskij. A lot of other results, remarks and conjectures on generic properties of finitely presented groups are contained in chapter 9 of M. Gromov’s work “Asymptotic invariants of infinite groups” [Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)]. The paper under review contains detailed proofs and a thorough introduction to combinatorial group theory, which includes the study of diagrams, small cancellation theory, Dehn’s algorithm, Dehn functions and the combinatorial Besicovich inequality.

MSC:
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
57M07 Topological methods in group theory
53C22 Geodesics in global differential geometry
20P05 Probabilistic methods in group theory
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