**Journal of Lie Theory
**

Vol. 8, No. 2, pp. 255-277 (1998)

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Automorphisms and quasi-conformal mappings of Heisenberg type groups

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P. E. Barbano

Department of Mathematics

Yale University

10 Hill House Avenue

New Haven CT 06520

barbano@jules.math.yale.edu

**Abstract:** The Lie algebras of trace-zero derivations of Heisenberg-type groups are explicitly characterized, along with the connected component of their groups of measure preserving automorphisms. We establish a general criterion on properties of the stabilizer of a lattice in a simply connected nilpotent Lie group and apply it to the full family of $H$-type Lie groups. A necessary condition for the existence of non-conformal quasi-conformal mappings on $H$-type groups is also given.

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