  In 1999 Chas and Sullivan discovered that the homology H_*(LX) of the space
of free loops on a closed oriented smooth manifold X has a rich algebraic
structure called string topology. They proved that H_*(LX) is naturally a
Batalin-Vilkovisky (BV) algebra. There are several conjectures connecting the
string topology BV algebra with algebraic structures on the Hochschild
cohomology of algebras related to the manifold X, but none of them has been
verified for manifolds of dimension n>1.
  In this work we study string topology in the case when X is aspherical (i.e.
its homotopy groups \pi_i(X) vanish for i > 1). In this case the Hochschild
cohomology Gerstenhaber algebra HH^*(A) of the group algebra A of the
fundamental group of X has a BV structure. Our main result is a theorem
establishing a natural isomorphism between the Hochschild cohomology BV algebra
HH^*(A) and the string topology BV algebra H_*(LX). In particular, for a closed
oriented surface X of hyperbolic type we obtain a complete description of the
BV algebra operations on H_*(LX) and HH^*(A) in terms of the Goldman bracket of
loops on X. The only manifolds for which the BV algebra structure on H_*(LX)
was known before were spheres and complex Stiefel manifolds.
  Our proof is based on a combination of topological and algebraic
constructions allowing us to compute and compare multiplications and BV
operators on both H_*(LX) and HH^*(A).
