  The m-thick part of the modular surface X is the smallest compact subsurface
of X with horocycle boundary containing all the closed geodesics which wind
around the cusp at most m times. The m-thick parts form a compact exhaustion of
X. We are interested in the geodesics that lie in the m-thick part (so called m
low-lying geodesics). We produce a complete asymptotic expansion for the number
of m low-lying geodesics of length equal to 2n in the modular surface. In
particular, we obtain the asymptotic growth rate of the m low-lying geodesics
in terms of their word length using the natural generators of the modular
group. After establishing a correspondence between this counting problem and
the problem of counting necklaces with n beads, we perform a careful
singularity analysis on the associated generating function of the sequence.
