We investigate maximal tori in the Hochschild cohomology Lie
algebra HH1
(A) of a finite dimensional algebra A, and their connection with
the fundamental groups associated to presentations of A. We prove that every
maximal torus in HH1
(A) arises as the dual of some fundamental group of
A, extending work of Farkas, Green and Marcos; de la Pe˜na and Saor´ın; and
Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental
group of A is a derived invariant quantity, and among self-injective algebras,
an invariant under stable equivalences of Morita type. Using this we prove
that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for
very restricted classes of monomial algebras.