Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group

By Yehuda Shalom. 2000

170 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/41b982cd39ca17ef4180b72dd809de57db1ed88c

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Isometric Group Actions on Hilbert Spaces: Growth of Cocycles (2005, background): “then for some constant C>0, kF(x)k2 = Cd(x,x 0)+o(d(x,x0)). Set σ= π|Γ; then kb|Γ(g)k2 ∼ d(gx0,x0) |g|S. Let us prove that b|Γ is not an almost coboundary. First, we appeal to a result of Shalom [Sha00], stating that the restriction map H1(G,π) → H1(Γ,σ) 6The proof would be simpler in that it wo”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, background): “7 in [38]) shows that Γ < Sp(n, 1) is integrable, thus 1-measure equivalent to Λ.”
Sparse equidistribution problems, period bounds and subconvexity (2005, background): “1Cjxjv/ ̨ : This (or an even stronger form) is stated and used in [39]; in the present case it can be verified by direct computation.”
Factor and normal subgroup theorems for lattices in products of groups (2006, background): “... in [Sha00a] it is shown that Theorem 1.2 does hold also in the noncocompact case, provided a certain “square-integrability” (hereafter abbreviated S-I) condition holds for some (cocycle defined on a) fundamental domain of Γ. This condition makes sense for an arbitrary finitely generated lattice ”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “3] for n = 3 (based on an earlier idea of [60]), and its proof easily extends to general n ≥ 2.”
Complex hyperbolic triangle groups (2003, background): “ understanding triangle groups generated by real reﬂections. See [3] and [4]. There has been a lot of other great work done recently on complex hyperbolic discrete groups, for instance [1], [2], [9], [20], [21]. Also see the references in Goldman’s book [f8]. 5. A conjectural picture We will conside”
Strong solidity of group factors from lattices in SO(n,1) and SU(n,1) (2010, methodology): “9 in [27] shows that Γ possesses an unbounded cocycle into some strongly l representation for p ≥ 2.”
Integrable measure equivalence and rigidity of hyperbolic lattices (2010, unknown): “§4]), and in Shalom’s [55].”
Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, background): “(We refer to [42] for the arguments below.)”
Groups acting on manifolds: around the Zimmer program (2008, result): “r the case of SU(1,n). If this is true, it immediately implies that actions of those lattices can never be classiﬁed. There are still some interesting results concerning actions of these lattices. In [208], Shalom places restrictions on the possible actions of SO(1,n) and SU(1,n) in terms of the fun”
Real hyperbolic on the outside, complex hyperbolic on the inside (2003, background): “For a representation-theoretic comparison of discrete subgroups in the different rank one spaces, see [ Sh ].”