Limiting modular symbols and their fractal geometry

Details Limiting modular symbols and their fractal geometry Limiting modular symbols and their fractal geometry

2006-11-02

arxiv.org/abs/math/0611048v1

Homology at infinity; fractal geometry of limiting symbols for modular group, Topology 46 (2007), no. 5, 469-49

Mathematics

Geometric Topology

20 pages, 1 figure

Scientific paper

10.1016/j.top.2007.03.004

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an `almost complete' multifractal description of the higher--dimensional level sets arising from Manin--Marcolli's limiting modular symbols.

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