In this paper we study the dynamics of properly discontinuous and crystallographic affine semigroups leaving a hyperbolic form, i. e. a quadratic from of signature (n, 1) invariant. The motivating question here is a question stated by H. Abels, G. Margulis and the author: Is the Zariski closure of a crystallographic affine semigroup leaving a hyperbolic form invariant a virtually solvable group?We proved that that for n = 2 the answer is "yes".
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Acknowledgment. I thank H. Abels and G. Margulis for arousing my interest in this question and their collaboration. I also thank the referee for essential suggestions and important remarks. The author would like to thank several institutions and foundations for their support: SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”, the USA–Israel Binational Science Foundation under BSF grant 2004010, the Emmy Noether Research Institute for Mathematics, Bar Ilan University and the Israel Science Foundation under ISF grant 657/09 and the Max Planck Institute for Mathematics, Bonn.