The Teichm\"uller space of the standard action of $SL(2,Z)$ on $T^2$ is trivial

Abstract

The group $SL(n,{\bf Z})$ acts linearly on $\R^n$, preserving the integer lattice $\Z^{n} \subset \R^{n}$. The induced (left) action on the n-torus $\T^{n} = \R^{n}/\Z^{n}$ will be referred to as the ``standard action''. It has recently been shown that the standard action of $SL(n,\Z)$ on $\T^n$, for $n \geq 3$, is both topologically and smoothly rigid. That is, nearby actions in the space of representations of $SL(n,\Z)$ into ${\rm Diff}^{+}(\T^{n})$ are smoothly conjugate to the standard action. In fact, this rigidity persists for the standard action of a subgroup of finite index. On the other hand, while the $\Z$ action on $\T^{n}$ defined by a single hyperbolic element of $SL(n,\Z)$ is topologically rigid, an infinite dimensional space of smooth conjugacy classes occur in a neighborhood of the linear action. The standard action of $SL(2, \Z)$ on $\T^2$ forms an intermediate case, with different rigidity properties from either extreme. One can construct continuous deformations of the standard action to obtain an (arbritrarily near) action to which it is not topologically conjugate. The purpose of the present paper is to show that if a nearby action, or more generally, an action with some mild Anosov properties, is conjugate to the standard action of $SL(2, \Z)$ on $\T^2$ by a homeomorphism $h$, then $h$ is smooth. In fact, it will be shown that this rigidity holds for any non-cyclic subgroup of $SL(2, \Z)$.