By Ollivier Y.
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Additional resources for A January invitation to random groups
But, just as usual small cancellation stops at density 1/12 for random groups, relative small cancellation is too restrictive and does not make it up to the maximal number of elements one can kill, hence the interest of the random point of view. b. Growth of random quotients. e. for the definition 50 A January 2005 invitation to random groups of the growth exponent). Note that by the results in [AL02], this exponent cannot stay unchanged. Theorem 39 – Let G0 be a non-elementary, torsion-free hyperbolic group generated by the finite set S.
It is possible to prove [Zuk03] quite the same hyperbolicity theorem as for the density model: Theorem 29 – If d < 1/2, then with overwhelming probability a random group in the triangular model, at density d, is non-elementary hyperbolic. If d > 1/2, it is trivial with overwhelming probability. 42 A January 2005 invitation to random groups But the fact that groups in the triangular model are “larger”than those in the density model is especially clear when considering the following proposition.
Reusing the methods of Arzhantseva and Ol’shanski˘ı, Kapovich and Schupp prove that there is “only one” m-tuple generating the group. Recall [LS77] that for a m-tuple of elements (g1 , . . , gm ) in a group, a Nielsen move consists in replacing some gi with its inverse, or interchanging two gi ’s, or replacing some gi with gi gj for some i = j. Obviously these moves do not change the subgroup generated by the m-tuple. t. which the random presentation was taken. In particular, any automorphism of G lifts to an automorphism of Fm .