76,586 research outputs found
Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets
We present a proof of Rider's unpublished result that the union of two Sidon
sets in the dual of a non-commutative compact group is Sidon, and that randomly
Sidon sets are Sidon. Most likely this proof is essentially the one announced
by Rider and communicated in a letter to the author around 1979 (lost by him
since then). The key fact is a spectral gap property with respect to certain
representations of the unitary groups that holds uniformly over . The
proof crucially uses Weyl's character formulae. We survey the results that we
obtained 30 years ago using Rider's unpublished results. Using a recent
different approach valid for certain orthonormal systems of matrix valued
functions, we give a new proof of the spectral gap property that is required to
show that the union of two Sidon sets is Sidon. The latter proof yields a
rather good quantitative estimate. Several related results are discussed with
possible applications to random matrix theory.Comment: v2: minor corrections, v3 more minor corrections v4) minor
corrections, last section removed to be included in another paper in
preparation with E. Breuillard v5) more minor corrections + two references
added. The paper will appear in a volume dedicated to the memory of V. P.
Havi
Singularities of free group character varieties
Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued
representations of a rank r free group. We classify the algebraic singular
stratification of X. This comes down to showing that the singular locus
corresponds exactly to reducible representations if there exist singularities
at all. Then by relating algebraic singularities to topological singularities,
we show the moduli spaces X generally are not topological manifolds, except for
a few examples we explicitly describe.Comment: 33 pages. Version 4 is shorter and more focused; cut material will be
expanded upon and written up in subsequent papers. Clarifications, and
expository revisions have been added. Accepted for publication in Pacific
Journal of Mathematic
Dynamics of the mapping class group action on the variety of PSL(2,C) characters
We study the action of the mapping class group Mod(S) on the boundary dQ of
quasifuchsian space Q. Among other results, Mod(S) is shown to be topologically
transitive on the subset C in dQ of manifolds without a conformally compact
end. We also prove that any open subset of the character variety
X(pi_1(S),SL(2,C)) intersecting dQ does not admit a nonconstant
Mod(S)-invariant meromorphic function. This is related to a question of
Goldman.Comment: This is the version published by Geometry & Topology on 11 July 200
Wave Front Sets of Reductive Lie Group Representations II
In this paper it is shown that the wave front set of a direct integral of
singular, irreducible representations of a real, reductive algebraic group is
contained in the singular set. Combining this result with the results of the
first paper in this series, the author obtains asymptotic results on the
occurrence of tempered representations in induction and restriction problems
for real, reductive algebraic groups.Comment: Accepted to Transactions of the American Mathematical Societ
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