Limiting modular symbols and their fractal geometry

TL;DR

This paper explores the fractal geometry of boundary components in the homology of modular surface coverings, providing a symbolic dynamic representation and a multifractal analysis of limiting modular symbols.

Contribution

It introduces a symbolic dynamic model for geodesic flows on modular surfaces and offers a multifractal description of limiting modular symbols, revealing new boundary structure insights.

Findings

Complete symbolic representation of geodesic dynamics

Multifractal description of higher-dimensional level sets

Identification of algebraically invisible homology parts

Abstract

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.