Sofic groups and diophantine approximation

We prove the algebraic eigenvalue conjecture of J. Dodziuk, P. Linnell, V. Mathai, T. Schick, and S. Yates (see [2]) for sofic groups. Moreover, we give restrictions on the spectral measure of elements in the integral group ring. Finally, we define integer operators and prove a quantization of the operator norm below 2. To the knowledge of the author, there is no group known that is not sofic. (c) 2007 Wiley Periodicals, Inc.