Pattern Equivariant Representation Variety of Tiling Spaces for Any Group G

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.

💡 Research Summary

The paper introduces a novel topological invariant for tiling spaces, called the Pattern Equivariant Representation Variety (PREP). The authors begin by recalling the classical correspondence between flat connections on a trivial principal G‑bundle over a manifold M and the representation variety Hom(π₁(M), G)/G, where G is a connected Lie group. They then ask how this picture can be adapted to the highly non‑manifold setting of a tiling T of ℝᵈ, which is viewed as a marked copy of Euclidean space equipped with a finite set of prototiles and a substitution rule.

To capture the local combinatorial structure of T, the paper uses the standard approximant complexes Γₙ. Each Γₙ records the pattern of tiles that appear within a radius n of the origin; as n increases, the inverse limit of the Γₙ’s recovers the tiling space Ω_T. The key new notion is that of a pattern‑equivariant (PE) bundle over T. A PE bundle is a G‑principal bundle defined on each approximant Γₙ, with the additional requirement that whenever two points of Γₙ lie in regions that are locally indistinguishable (i.e., they have the same finite patch of tiles), the transition functions of the bundle agree. In other words, the bundle’s structure respects the tiling’s pattern equivalence relation.

The main theorem (Theorem 3.1) states that the set of isomorphism classes of PE bundles, denoted PREP(T), is naturally isomorphic to the direct limit

  limₙ Hom(π₁(Γₙ), G)/G,

where the bonding maps are induced by the natural cellular maps fₙ : Γₙ₊₁ → Γₙ. Crucially, the group G is allowed to be any group—finite, infinite, connected, disconnected, abelian or non‑abelian. This generality extends the classical picture far beyond Lie groups and shows that the construction is fundamentally combinatorial.

To illustrate the power of PREP, the authors compute the invariant for two well‑known one‑dimensional substitution tilings: the Period‑Doubling tiling and its double cover, the Thue‑Morse tiling. They take G = S₃, the symmetric group on three letters. For the Period‑Doubling tiling, each approximant Γₙ is a graph with two edges, whose fundamental group is a free group F₂. The representation set Hom(F₂, S₃)/S₃ consists of all homomorphisms from F₂ to S₃ modulo conjugation, which yields three distinct conjugacy classes. Passing to the direct limit leaves exactly three elements in PREP. For the Thue‑Morse tiling, the approximants are double covers of those for Period‑Doubling, introducing an extra relation that identifies two of the previously distinct classes. Consequently the direct limit contains only two elements. Thus PREP distinguishes the two tilings even though they share the same Čech cohomology and many other classical invariants.

The paper also discusses the relationship between PREP and more familiar invariants. While Čech cohomology captures additive information about the tiling space, PREP encodes non‑abelian data arising from the fundamental groups of the approximants and the way these groups map into an arbitrary target group G. Consequently, PREP can detect differences invisible to cohomology, K‑theory, or pattern‑equivariant cohomology. Moreover, because the bonding maps fₙ reflect the substitution rule, PREP is sensitive to the dynamical properties of the tiling, linking it to the study of tiling automorphisms and spectral theory.

In the concluding section, the authors outline several directions for future work. They suggest extending the computation to higher‑dimensional tilings such as Penrose or Ammann‑Beenker tilings, exploring algorithmic methods for handling infinite or non‑abelian groups G, and investigating connections with physical models (e.g., topological phases of matter) where pattern‑equivariant bundles naturally arise. They also propose studying the interaction between PREP and the tiling’s automorphism group, which could lead to a richer classification scheme for aperiodic order.

Overall, the paper provides a robust, group‑theoretic framework for extracting fine‑grained topological information from tiling spaces, demonstrating both theoretical depth and concrete applicability through explicit calculations.