A January 2005 invitation to random groups

Abstract

Let $X_m=\{x_1,\dots,x_m\}$, $m\ge 2$, be a finite set and let ${\scr Q_m}$ be the set of all pairs $(G,X_m)$, where $G$ is a group generated by $X_m$. A random group is a random representative of the space ${\scr Q_m}$ (or rather of the space ${\scr Q_m^n}$ of groups given by $n\ge 1$ relators). Study of such a group leads to a uniform vision on (finitely generated or finitely presented) groups in general. This point of view is due to Gromov and goes back to his seminal paper on hyperbolic groups [M. L. Gromov, in {\it Essays in group theory}, 75--263, Springer, New York, 1987; [msn] MR0919829 (89e:20070) [/msn]]. A second approach consists of asking questions about randomly selected elements of a given group: what is its expected order, what are subgroups generated by random elements, and so on. It has already been shown to be a powerful tool for understanding finite groups. Infinite random groups in combinatorial, statistical, and topological meanings have yet to be investigated fully. \par The text under review is a survey of the results on random groups and on typical elements in a given infinite group known by January 2005. It begins with a gentle introduction to the geometric group theory. This part is short, contains all necessary background, and is useful for readers outside of the field. The main section titles are: (I) Models of typical groups; (II) Typical elements in a group; (III) Applications: random ingredients in specific constructions; (IV) Open problems and perspectives; (V) Proof of the density one half theorem. \par A random group $G$ is defined as a group given by a presentation $G=\langle X_m \mid R\rangle$ by generators and relators taken at random. To choose a model of a random group $G$ is to specify a probability law for the set $R$ of relators. \par In (I) various statistical models of a finitely presented random group are discussed. Namely, a parameter relevant to the length of words in $R$ is chosen and typical group properties are studied when this parameter is large. Basically, three models appear in such a way: the few-relator model (the number of relators in $R$ is fixed and the parameter bounds the lengths of relators; this model appears already, for example, in [V. S. Guba, Izv. Vyssh. Uchebn. Zaved. Mat. {\bf 1986}, no.~7, 12--19, 87; [msn] MR0867603 (88d:20048) [/msn]]), the few-relator model with various lengths (the number of relators is fixed, the relators are allowed to have different lengths, and the parameter is the minimum of these lengths), and the density model (the density parameter provides a control on the number of relators in $R$). In all these models, a typical finitely presented group is hyperbolic. \par The author compares different models, presents possible variations of them (such as a triangular model), and surveys, in detail, known results. He focuses on results showing how various values of the density parameter influence the group structure. In particular, how it is related to the CAT(0)-ness and hyperbolicity, small cancellation properties, freeness of subgroups, growth, Kazhdan's property (T) and the Haagerup property. For instance, if density is less than 1/2, then the random group is very probably infinite hyperbolic, whereas it is trivial at densities above 1/2. \par Finally, the space of marked groups is defined. This is the space ${\scr Q_m}$ endowed with a natural topology. This yields the notion of a generic group (although an appropriate combination with previous statistical results is necessary). \par In (II) two main examples are given. The first is the robustness of hyperbolicity: typical elements in a torsion-free non-elementary hyperbolic group can be ``killed'' (that is, added as a new relator) without harming the group structure too much. That is, such a random quotient is again a non-elementary hyperbolic group. The second example is a sharp counting of the number of one-relator groups up to isomorphism based on a rigidity property of one-relator groups. \par In (III) two applications are discussed. The first is the construction of Gromov of a finitely presented group whose Cayley graph quasi-contains an infinite family of expanding graphs. A roadmap to this construction is given. In particular, a random graphical quotient model of a random group, due to Gromov, is explained. In this model, given a finite graph and a symmetric set of group generators, one considers a random labelling of the edges by generators. Then one takes the quotient of the group by new relators which are all words read on cycles of the graph. Note that a random group obtained in such a combinatorial way by the random graphical quotient method is no longer a typical representative of ${\scr Q_m}$. The second application is the construction of Kazhdan groups whose outer automorphism group contains an arbitrary countable group. \par In (IV) the author comments on many open problems and further perspectives of the theory of random groups. The reader has to pay attention to (IV.k), where a very promising, Gromov temperature model of a random group is considered. \par In (V), the author gives the proof of the above-mentioned 1/2 phase transition theorem. There are no other proofs in this text. \par To sum up, it seems that we are only beginning to understand serious applications of the fascinating theory of random groups. In this context, this survey will be helpful not only for specialists in geometric group theory but for everybody who wants to discover the precise meaning of ``random groups''.