Isospectral convex domains in the hyperbolic plane

We construct pairs of nonisometric convex polygons in the hyperbolic plane for which the Laplacians are both Dirichlet and Neumann isospectral. We also give examples of pairs of isospectral potentials for the Schrodinger operator on certain convex hyperbolic polygons. Given a bounded domain Ql with piecewise-smooth boundary in a Riemanman manifold M, denote by SpecD(Q) (respectively, SpecN(fl)) the eigenvalue spectrum of the Laplace-Beltrami operator acting on smooth functions on Ql with Dinchlet (respectively, Neumann) boundary conditions. A pair of domains lI and Q2 in .M is Dirichlet isospectral if SpecD(Q1) = SpecD(02); Neumann isospectrality is similarly defined. Mark Kac's question "Can one hear the shape of a drum?" [K] asks whether Dinchlet isospectral domains in the Euclidean plane must be isometric. Recently, the authors and Wolpert [GWW1, 2] answered Kac's question negatively by exhibiting a pair of nonisometric domains in the Euclidean plane which are Dinchlet and Neumann isospectral; the construction also yields isospectral domains in the round 2-sphere and in the hyperbolic plane. Using similar methods, Buser et al. [BCDS] constructed other examples. In all cases, however, the domains are nonconvex. Thus Kac's question for convex plane domains remains open. The purpose of this note is to exhibit pairs of convex domains in the hyperbolic plane which are both Dinchlet and Neumann isospectral. These domains are obtained from those of [GWW 1, 2] by modifying the shape of the fundamental tile used in the construction, as in [BCDS]; Berard's extension [BI] of Sunada's Theorem [S] facilitates the proof of isospectrality by "transplantation" of eigenfunctions from one domain to the other, as in [BCDS, B2]. Let T be a hyperbolic triangle with vertex angles a, ,8, and y, and let