arXiv subjects
Publications and source records attributed to Yehuda Shalom.
We prove that every infinite subgroup of a higher rank irreducible lattice in a semi-simple Lie group has finite index in the lattice, proving the Margulis-Zimmer conjecture for these lattices. The method of proof combines the method used in a work of Shalom and Willis and also some work of Raghunathan and Venkataramana.
Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.
In this paper we introduce for a group $G$ the notion of ultralimit of measure class preserving actions of it, and show that its Furstenberg-Poisson boundaries can be obtained as an ultralimit of actions on itself, when equipped with appropriately chosen measures. We use this result in embarking on a systematic quantitative study of the basic question how close to invariant one can find measures on a $G$-space, particularly for the action of the group on itself. As applications we show that on amenable groups there are always "almost invariant measures" with respect to the information theoretic Kullback-Leibler divergence (and more generally, any $f$-divergence), making use of the existence of measures with trivial boundary. More interestingly, for a free group $F$ and a symmetric measure $λ$ supported on its generators, one can compute explicitly the infimum over all measures $η$ on $F$ of the Furstenberg entropy $h_λ(F,η)$. Somewhat surprisingly, while in the case of the uniform measure on the generators the value is the same as the Furstenberg entropy of the Furstenberg-Poisson boundary of the same measure $λ$, in general it is the Furstenberg entropy of the Furstenberg-Poisson boundary of a measure on $F$ different from $λ$.
We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy of Margulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan's property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.
We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $Γ,Λ$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action $Γ\curvearrowright G/Λ$ is rigid. If in addition $G$ has property (T) then we derive that the von Neumann algebra $L^{\infty}(G/Λ)\rtimesΓ$ has property (T). We also show that if the adjoint action of $G$ on the Lie algebra of $G$ - $\{0\}$ is amenable (e.g. if $G=SL_2(\Bbb R)$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $Γ\curvearrowright G/Λ$ is either hyperfinite or rigid.
We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d >0$: If there is some $ R_0 > \exp(\exp(Cd^C))$ for which the number of elements in a ball of radius $R_0$ in a Cayley graph of $G$ is bounded by $R_0^d$, then $G$ has a finite index subgroup which is nilpotent (of step $<C^d$). An effective bound on the finite index is provided if "nilpotent" is replaced by 'polycyclic", thus yielding a non-trivial result for finite groups as well.
Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.
We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type II_1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.
The article establishes a long list of rigidity properties of lattices in G = SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity of lattices in higher-rank Lie groups. The arguments are set in the context of strongly L^p unitary representations of G.