By Ollivier Y.

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The drawback is that these techniques only work for very small densities. 3. Counting one-relator groups On a very different topic, consideration of generic-case rather than worstcase behavior for algorithmic problems in group theory (most notably the isomorphism problem) led I. Kapovich, Myasnikov, Schupp and Shpilrain, in a series of closely related papers [KSS, KS, KS05, KMSS05, KMSS03], to the conclusion that generic elements are often nicely behaved. The frontier between properties of one-relator groups and properties of a typical word in the free group is faint; for this review we selected an application where the emphasis is really put on the group, namely, evaluation of the number of distinct one-relator groups up to isomorphism.

Am | R where R is a random set of relators at density d, at length . We say that a property of R, or of G, occurs with overwhelming probability at density d if its probability of occurrence tends to 1 as → ∞, for fixed d. g. replacing the sphere S with the ball B of words of length at most . c. Of course, the main point in this definition is the number (2m − 1)d of relators taken, which is actually quite large. ). g. for the dimension of an intersection). The basic idea is that d is the “dimension” of the random set of relators R (the set S itself being considered of dimension because we have independent letter choices to make to specify an element in S ).

We can either take the integer part, or choose two constants C1 and C2 and consider taking any number of words between C1 (2m − 1)d and C2 (2m − 1)d . Once more this does not affect our statements at all. One may hesitate between choosing reduced or cyclically reduced words. Once again this does not matter. Section 4 of [Oll04] (in particular Remark 8) contains an axiomatic framework which allows to handle such a loose model and not to reprove all the Models of typical groups 33 theorems for each variant.

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A January invitation to random groups by Ollivier Y.

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