PSLn(q) as operator group of isospectral drums

Abstract

In a paper by M Kac (1966 Am. Math. Mon. 73 1–23), Kac asked his famous question ‘Can one hear the shape of a drum?’, which was answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. In Buser et al (1994 Int. Math. Res. Not. 9), it is observed that all operator groups associated with the known counter examples are isomorphic to one of PSL3(2), PSL3(3), PSL4(2) and PSL3(4). We show that if (D1, D2) is a pair of non-congruent planar isospectral domains constructed from unfolding a polygonal base-tile and with associated operator group PSLn(q), then (n, q) belongs to this very restricted list.