Representation equivalent Bieberbach groups and strongly isospectral flat manifolds

Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and $\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\Gamma_1\backslash G)$ and $L^2(\Gamma_2\backslash G)$ 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 $\mathcal O$ in \F=\R,\C or $\mathbb{H}$, and an \F-hermitian form $Q$ of signature $(n,1)$ with coefficients in $\mathcal O$. Let $k\in\N$. By applying a lattice point theorem on the \F-hyperbolic space, we give an asymptotic formula with an error term, as $t\to+\infty$, for the number $N_t(Q,-k)$ of integral solutions $x\in\mathcal O^{n+1}$ of the equation $Q[x]=-k$ satisfying $|x_{n+1}|\leq t$.Comment: To appear in Proceedings of the American Mathematical Societ

Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of $G$. In this paper we study whether the functional $g\mapsto \lambda_1(G/K,g)\operatorname{diam}(G/K,g)^2$ is bounded among $G$-invariant metrics $g$ on $G/K$. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when $K$ is trivial; the only particular cases known so far are when $G$ is abelian, $\operatorname{SU}(2)$, and $\operatorname{SO}(3)$. In this article we prove the existence of the mentioned upper bound for every compact homogeneous space $G/K$ 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 $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is K\"ahler while $M'$ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for $n=24$ and $k=3$ 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 $n$-dimensional lens space $L$, we associate a congruence lattice $\mathcal L$ in $\mathbb Z^m$, with $n=2m-1$ and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on $L$ with the number of lattice elements of a given $\|\cdot\|_1$-length in $\mathcal L$. As a consequence, we show that two lens spaces are isospectral on functions (resp.\ isospectral on $p$-forms for every $p$) if and only if the associated congruence lattices are $\|\cdot\|_1$-isospectral (resp.\ $\|\cdot\|_1$-isospectral plus a geometric condition). Using this fact, we give, for every dimension $n\ge 5$, infinitely many examples of Riemannian manifolds that are isospectral on every level $p$ and are not strongly isospectral.Comment: Accepted for publication in IMR