Combinatorial growth in the modular group. (English) Zbl 1512.20154
Summary: We consider an exhaustion of the modular orbifold by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so named low lying reciprocal geodesics) converges to the growth rate of the full set of reciprocal geodesics on the modular orbifold. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics.
MSC:
| 20F69 | Asymptotic properties of groups |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 53C22 | Geodesics in global differential geometry |
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