360 research outputs found

### Twisted limit formula for torsion and cyclic base change

Let $G$ be the group of complex points of a real semi-simple Lie group whose
fundamental rank is equal to 1, e.g. G= \SL_2 (\C) \times \SL_2 (\C) or
\SL_3 (\C). Then the fundamental rank of $G$ is $2,$ and according to the
conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the
very weak sense of 'subexponential in the co-volume' --- torsion homology.
Using base change, we exhibit sequences of lattices where the torsion homology
grows exponentially with the \emph{square root} of the volume.
This is deduced from a general theorem that compares twisted and untwisted
$L^2$-torsions in the general base-change situation. This also makes uses of a
precise equivariant 'Cheeger-M\"uller Theorem' proved by the second author
\cite{Lip1}.Comment: 23 page

### Tetrahedra of flags, volume and homology of SL(3)

In the paper we define a "volume" for simplicial complexes of flag
tetrahedra. This generalizes and unifies the classical volume of hyperbolic
manifolds and the volume of CR tetrahedra complexes. We describe when this
volume belongs to the Bloch group. In doing so, we recover and generalize
results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to
the work of Fock and Goncharov.Comment: 45 pages, 14 figures. The first version of the paper contained a
mistake which is correct here. Hopefully the relation between the works of
Neumann-Zagier on one side and Fock-Goncharov on the other side is now much
cleare

### Eigenfunctions and Random Waves in the Benjamini-Schramm limit

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on
Riemannian manifolds. We show that Benjamini-Schramm convergence provides a
unified language for the level and eigenvalue aspects of the theory. As a
result, we present a mathematically precise formulation of Berry's conjecture
for a compact negatively curved manifold and formulate a Berry-type conjecture
for sequences of locally symmetric spaces. We prove some weak versions of these
conjectures. Using ergodic theory, we also analyze the connections of these
conjectures to Quantum Unique Ergodicity.Comment: 40 page

### The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)

In Part 1 of this paper we construct a spectral sequence converging to the
relative Lie algebra cohomology associated to the action of any subgroup $G$ of
the symplectic group on the polynomial Fock model of the Weil representation,
see Section 7. These relative Lie algebra cohomology groups are of interest
because they map to the cohomology of suitable arithmetic quotients of the
symmetric space $G/K$ of $G$. We apply this spectral sequence to the case $G =
\mathrm{SO}_0(n,1)$ in Sections 8, 9, and 10 to compute the relative Lie
algebra cohomology groups $H^{\bullet} \big(\mathfrak{so}(n,1), \mathrm{SO}(n);
\mathcal{P}(V^k) \big)$. Here $V = \mathbb{R}^{n,1}$ is Minkowski space and
$\mathcal{P}(V^k)$ is the subspace of $L^2(V^k)$ consisting of all products of
polynomials with the Gaussian. In Part 2 of this paper we compute the
cohomology groups $H^{\bullet}\big(\mathfrak{so}(n,1), \mathrm{SO}(n); L^2(V^k)
\big)$ using spectral theory and representation theory. In Part 3 of this paper
we compute the maps between the polynomial Fock and $L^2$ cohomology groups
induced by the inclusions $\mathcal{P}(V^k) \subset L^2(V^k)$.Comment: 64 pages, 5 figure

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