A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups |
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Authors: | Lewis Bowen |
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Institution: | (1) Mathematics, Indiana University, Rawles Hall, Indiana Univ, Bloomington, IN 47405, USA |
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Abstract: | The classical prime geodesic theorem (PGT) gives an asymptotic formula (as x tends to infinity) for the number of closed geodesics with length at most x on a hyperbolic manifold M. Closed geodesics correspond to conjugacy classes of π1(M) = Γ where Γ is a lattice in G = SO(n,1). The theorem can be rephrased in the following format. Let be the space of representations of into Γ modulo conjugation by Γ. is defined similarly. Let be the projection map. The PGT provides a volume form vol on such that for sequences of subsets {B
t
}, satisfying certain explicit hypotheses, |π−1(B
t
)| is asymptotic to vol(B
t
). We prove a statement having a similar format in which is replaced by a free group of finite rank under the additional hypothesis that n = 2 or 3.
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Keywords: | Subgroup growth Prime geodesic theorem Free subgroup Character variety Hyperbolic group Hyperbolic geometry |
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