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Description

Ever since Mark Kac's important paper ``Can one hear the shape of the drum?'' there has been interest in finding compact Riemannian manifolds $X$ and $Y$ which are isospectral (i.e. the spectrum of the Laplacian, $\Delta$, acting on $L^2(X)$ is equal to the spectrum of $\Delta$ acting on $L^2(Y)$), but not isomorphic. Most known isospectral pairs which are not isomorphic are commensurable (i.e. they share a common finite cover). For $d \geq 3$, $G = PGL_d(F)$, $K$, a maximal compact subgroup of $G$, $S= G/K$ and for any $m \in \N$, we will present a family of $m$ co-compact arithmetic lattices $\{\Gamma_i\}$ in $G$ such that $\Gamma_i \backslash S$ are isospectral and not commensurable. Here $F = \R, \C$, or a local field of positive characteristic.

The construction is based on arithmetic groups obtained from division algebras with the same ramification points, but with different invariants. The proof transfers the problem to representation theory and uses the Jacquet-Langlands Correspondence.