26 research outputs found
Representation equivalent Bieberbach groups and strongly isospectral flat manifolds
Let and be Bieberbach groups contained in the full
isometry group of . We prove that if the compact flat
manifolds and
are strongly isospectral then the Bieberbach groups and
are representation equivalent, that is, the right regular representations
and are unitarily
equivalent.Comment: To appear in Canadian Mathematical Bulleti
An asymptotic formula for representations of integers by indefinite hermitian forms
We fix a maximal order in \F=\R,\C or , and an
\F-hermitian form of signature with coefficients in .
Let . By applying a lattice point theorem on the \F-hyperbolic space,
we give an asymptotic formula with an error term, as , for the
number of integral solutions of the equation
satisfying .Comment: To appear in Proceedings of the American Mathematical Societ
Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds
Let be a compact connected Lie group and let be a closed subgroup of
. In this paper we study whether the functional is bounded among -invariant
metrics on . Eldredge, Gordina, and Saloff-Coste conjectured in 2018
that this assertion holds when is trivial; the only particular cases known
so far are when is abelian, , and
. In this article we prove the existence of the mentioned
upper bound for every compact homogeneous space having multiplicity-free
isotropy representation.Comment: Accepted for publication in Transformation Groups. arXiv admin note:
text overlap with arXiv:2004.0035
Strongly isospectral manifolds with nonisomorphic cohomology rings
For any , , we give pairs of compact flat -manifolds with holonomy groups , that are strongly isospectral, hence
isospectral on -forms for all values of , having nonisomorphic cohomology
rings. Moreover, if is even, is K\"ahler while is not.
Furthermore, with the help of a computer program we show the existence of large
Sunada isospectral families; for instance, for and there is a
family of eight compact flat manifolds (four of them K\"ahler) having very
different cohomology rings. In particular, the cardinalities of the sets of
primitive forms are different for all manifolds.Comment: 25 pages, to appear in Revista Matem\'atica Iberoamerican
Spectra of lens spaces from 1-norm spectra of congruence lattices
To every -dimensional lens space , we associate a congruence lattice
in , with and we prove a formula relating
the multiplicities of Hodge-Laplace eigenvalues on with the number of
lattice elements of a given -length in . As a
consequence, we show that two lens spaces are isospectral on functions (resp.\
isospectral on -forms for every ) if and only if the associated
congruence lattices are -isospectral (resp.\
-isospectral plus a geometric condition). Using this fact, we
give, for every dimension , infinitely many examples of Riemannian
manifolds that are isospectral on every level and are not strongly
isospectral.Comment: Accepted for publication in IMR