Trees, Amalgams and Calogero-Moser Spaces

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $A_1$. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key r\^ole in our approach is played by a transitive action of the automorphism group of the free algebra $\c < x, y >$ on the Calogero-Moser varieties $\CC_n$ defined in \cite{BW}. Our results generalize well-known theorems of Dixmier and Makar-Limanov on automorphisms of $A_1$, answering an old question of Stafford (see \cite{St}). Finally, we propose a natural extension of the Dixmier Conjecture for $A_1$ to the class of Morita equivalent algebras.