In the 1980s, Goldman introduced a Lie algebra structure on the free
vector space generated by the free homotopy classes of oriented closed
curves in any orientable surface F. This Lie bracket is known as the
Goldman bracket and the Lie algebra is known as the Goldman Lie algebra.
In this dissertation, we compute the center of the Goldman Lie algebra for
any hyperbolic surface of finite type. We use hyperbolic geometry and
geometric group theory to prove our theorems. We show that for any
hyperbolic surface of finite type, the center of the Goldman Lie algebra
is generated by closed curves which are either homotopically trivial or
homotopic to boundary components or punctures.
We use these results to identify the quotient of the Goldman Lie algebra
of a non-closed surface by its center as a sub-algebra of the first
Hochschild cohomology of the fundamental group.
Using hyperbolic geometry, we prove a special case of a theorem of Chas,
namely, the geometric intersection number between two simple closed
geodesics is the same as the number of terms (counted with multiplicity)
in the Goldman bracket between them.
We also construct infinitely many pairs of length equivalent curves in any
hyperbolic surface F of finite type. Our construction shows that given a
self-intersecting geodesic x of F and any self-intersection point P of x,
we get a sequence of such pairs.