##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Jeroen Shillewaert

#### University of Canterbury, New Zealand \\ Department of Mathematics and Statistics

## A group-theoretic characterization of known counterexamples to the planar Kac conjecture

##### Abstract:

We investigate pairs of Euclidean TI-domains which are isospectral but not congruent. For Riemannian manifolds there are several examples which are isospectral but not isometric, see e.g. J. Milnor [2]. The method we use dates back to T. Sunada [3] considering the problem as a geometric analogue of a method in number theory which uses Dedekind zeta functions. Counter examples to M. Kac’s conjecture so-far can all be constructed by a certain tiling method (“transplantability”) using special linear operator groups which act 2-transitively on certain associated modules. These can be represented by colored graphs, which yield information on the fixpoint structure of the groups. It is shown that if any such operator group acts 2-transitively onthe associated module, no new counter examples can occur.\\ \footnotesize \noindent [1] M. KAC. Can one hear the shape of the drum?, Amer. Math. Monthly 73 (4, part 2) (1966), 1–23. \\ \noindent [2] J. MILNOR. Eigenvalues of the Laplace operators on certain manifolds, Proc. Nat. Acad. Sci. USA 51 (1964), 542. \\ \noindent [3] T. SUNADA. Riemannian Coverings and Isospectral Manifolds, Ann. Math. 121 (1980), 169–186.

Host: Jeff Remmel

### September 29, 2009

### 4:00 PM

### AP&M 7321

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