We establish a new spectral criterion for Kazhdan's property (T ) which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property (T ) for the groups ELn(R), where n ≥ 3 and R is an arbitrary finitely generated associative ring. We also strengthen some of the results on property (T ) for Kac-Moody groups from [DJ].

Abstract. In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group G is a group Comm(G) which depends only on the commensurability class of G. We study various properties of Comm(G); in particular, we find two natural ways to turn it into a topological group. We also use Comm(G) to study topological groups which contain G as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, such as Pink's analogue of Mostow's strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups.

Abstract. In this paper we introduce the concept of weighted deficiency for abstract and pro-p groups and study groups of positive weighted deficiency which generalize Golod-Shafarevich groups. In order to study weighted deficiency we introduce weighted versions of the notions of rank for groups and index for subgroups and establish weighted analogues of several classical results in combinatorial group theory, including the Schreier index formula.Two main applications of groups of positive weighted deficiency are given. First we construct infinite finitely generated residually finite p-torsion groups in which every finitely generated subgroup is either finite or of finite index -these groups can be thought of as residually finite analogues of Tarski monsters. Second we develop a new method for constructing just-infinite groups (abstract or pro-p) with prescribed properties; in particular, we show that graded group algebras of just-infinite groups can have exponential growth. We also prove that every group of positive weighted deficiency has a hereditarily just-infinite quotient. This disproves a conjecture of Boston on the structure of quotients of certain Galois groups and solves Problem 15.18 from Kourovka notebook.

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In this paper we survey the main results about Golod–Shafarevich groups and their applications in algebra, number theory and topology.