In 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 talk, 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 algebraof 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 aself-intersecting geodesic x of
F and any
self-intersection point P of x,we get a sequence of such pairs.