Pattern Equivariant Representation Variety of Tiling Spaces for Any Group G

TL;DR

This paper introduces a new topological invariant called PREP(T) for tiling spaces, generalizing representation varieties, and demonstrates its ability to distinguish between different tilings like Period Doubling and Thue-Morse.

Contribution

It defines the pattern equivariant bundle and invariant PREP(T), linking it to representation varieties and providing a method to distinguish tilings.

Findings

PREP(T) is isomorphic to a direct limit of representation varieties.

The invariant can distinguish between Period Doubling and Thue-Morse tilings.

Explicit calculations for G = S_3 demonstrate the invariant's effectiveness.

Abstract

It is well known that the moduli space of flat connections on a trivial principal bundle MxG, where G is a connected Lie group, is isomorphic to the representation variety Hom(\pi_1(M), G)/G. For a tiling T, viewed as a marked copy of R^d, we define a new kind of bundle called pattern equivariant bundle over T and consider the set of all such bundles. This is a topological invariant of the tiling space induced by T, which we call PREP(T), and we show that it is isomorphic to the direct limit lim_{f_n} Hom(\pi_1(\Gamma_n), G)/G, where \Gamma_n are the approximants to the tiling space and f_n are maps between them. G can be any group. As an example, we choose G to be the symmetric group S_3 and we calculate this direct limit for the Period Doubling tiling and its double cover, the Thue-Morse tiling, obtaining different results. This is the simplest topological invariant that can distinguish these two examples.