Alexander Lubotzky

Explicit expanders and the Ramanujan conjectures

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Powerful p-groups. I. Finite groups

In this paper we study a special class of finite p-groups, which we call powerful p-groups. In the second part of this paper, we apply our results to the study of p-adic analytic groups. This application is possible, because a finitely generated pro-p group is p-adic analytic if and only if it is “virtually pro-powerful.” These applications are described in the introduction to the second part, while now we describe the present part in more detail. In the first section we define a powerful p-group, as one whose subgroup of pth powers contains the commutator subgroup. We give several results on these groups, in particular show that many naturally defined subgroups of them are also powerful, and then use this to show that if H is any subgroup of the powerful group G, then the number of generators of H is at most the same number for G. This result and its “converse” can be regarded as one of our main results, the said converse stating that, if all subgroups of the p-group G can be generated by at most Y elements, then G contains a powerful subgroup whose index is bounded by a function of r only. Thus the study of powerful groups is related to the study of “groups of rank r,” i.e., groups with bound r on the number of generators of subgroups, as above. This connection is exploited in Section 2, where we use it, e.g., to establish a conjecture of Jones and Wiegold, that the number of generators of the multiplicator of a group of rank r is bounded by a function of r. Section 3 contains examples and some further results. Some of these relate our results to concepts from the “power-structure” of p-groups, as discussed in a previous paper by the second author [Ma]. There is some difference, both in the definitions and in the results, between the odd primes and the prime 2. Thus, in the first three sections we assume that p is an odd prime, while in the last section we take p = 2. In that section we just repeat the statement of the results of the previous sections, as far as we know them to be true, with the necessary modifications. We give proofs only if they differ from the ones for odd primes. The result corresponding to 1.8, e.g., is numbered 4.1.8, etc. 484 OC21-8693/87

The probability of generating a finite classical group

3.00

Expander graphs in pure and applied mathematics

Two randomly chosen elements of a finite simple classical group G are shown to generate G with probability →1 as ‖G‖ → ∞. Extensions of this result are presented, along with applications to profinite groups.

The word and Riemannian metrics on lattices of semisimple groups

Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms, and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry, and more. This expository article describes their constructions and various applications in pure and applied mathematics.

Rank-1 phenomena for mapping class groups

Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.

Subgroup growth and congruence subgroups

Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the Teichmuller space T (Σg) to lattices in simple Lie groups and their actions on the associated symmetric spaces. Indeed, the groups Mod(Σg) share many of the properties of (arithmetic) lattices in semisimple Lie groups. For example they satisfy the Tits alternative, they have finite virtual cohomological dimension, they are residually finite, and each of their solvable subgroups is polycyclic. A well-known dichotomy among the lattices in simple Lie groups is between lattices in rank one groups and higher-rank lattices, i.e. those lattices in simple Lie groups of R-rank at least two. It is somewhat mysterious whether Mod(Σg) is similar to the former or the latter. Some higher rank behavior of Mod(Σg) is indicated by the cusp structure of moduli space, by the fact that Mod(Σg) has Serre’s property (FA) [CV], and by Ivanov’s version (see, e.g. [Iv2]) for Mod(Σg) of Tits’s Theorem on automorphism groups of higher rank buildings. In this note we add two more properties to the list (see, e.g. [Iv1, Iv2, Iv3] and the references therein) of properties which exhibits similarities of Mod(Σg) with lattices in rank one groups: every infinite order element of Mod(Σg) has linear growth in the word metric, and Mod(Σg) is not bound∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship. †Supported in part by the US-Israel BSF grant. ‡Supported in part by NSF grant DMS 9971596

The product replacement algorithm and Kazhdan’s property (T)

SummaryLetk be a global field,O its ring of integers,G an almost simple, simply connected, connected algebraic subgroups ofGLm, defined overk and Γ=G(O) which is assumed to be infinite. Let σn(Г) (resp. γn(Г) be the number of all (resp. congruence) subgroups of index at mostn in Γ. We show:(a) If char(k)=0 then:(i)

Explicit constructions of Ramanujan complexes of type A d

C_1 \frac{{\log ^2 n}}{{\log \log n}} \leqq \log \gamma _n (\Gamma ) \leqq C_2 \frac{{\log ^2 n}}{{\log \log n}}