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Bratteli diagrams, translation flows and their $C^*$-algebras

Ian F. Putnam, Rodrigo Treviño

TL;DR

The paper builds a detailed bridge between bi-infinite ordered Bratteli diagrams and translation surfaces, showing how path-space data, tail-equivalence, and carefully chosen orders yield a translation surface $S_{\mathcal{B}}$ together with an AF-type groupoid $T^{+}(X_{\mathcal{B}})$ and a foliation groupoid $\mathcal{F}^{+}(S_{\mathcal{B}})$. It develops a comprehensive C$^*$-algebra framework linking tail-equivalence algebras to foliation algebras, and computes their K-theory, establishing how finite vs infinite genus affects $K_0$-groups and the presence of torsion or rank. The work specializes to Chamanara’s infinite-genus surface and to finite-genus cases via Veech’s zippered rectangles and Rauzy–Veech/RH inductions, showing how the combinatorial data encode geometric and dynamical information. The results yield explicit K-theory invariants for a broad class of translation surfaces and expose a robust, computable pathway from Bratteli diagrams to flat geometry and operator algebras, with Chamanara’s and finite-genus examples illustrating the theory’s reach and limits.

Abstract

In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the $C^{*}$-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the $C^{*}$-algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved.

Bratteli diagrams, translation flows and their $C^*$-algebras

TL;DR

The paper builds a detailed bridge between bi-infinite ordered Bratteli diagrams and translation surfaces, showing how path-space data, tail-equivalence, and carefully chosen orders yield a translation surface together with an AF-type groupoid and a foliation groupoid . It develops a comprehensive C-algebra framework linking tail-equivalence algebras to foliation algebras, and computes their K-theory, establishing how finite vs infinite genus affects -groups and the presence of torsion or rank. The work specializes to Chamanara’s infinite-genus surface and to finite-genus cases via Veech’s zippered rectangles and Rauzy–Veech/RH inductions, showing how the combinatorial data encode geometric and dynamical information. The results yield explicit K-theory invariants for a broad class of translation surfaces and expose a robust, computable pathway from Bratteli diagrams to flat geometry and operator algebras, with Chamanara’s and finite-genus examples illustrating the theory’s reach and limits.

Abstract

In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the -algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the -algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved.
Paper Structure (27 sections, 106 theorems, 239 equations, 10 figures)

This paper contains 27 sections, 106 theorems, 239 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Bratteli diagrams: ordered and bi-infinite
  3. The path space
  4. Orders on the path space
  5. Singular points
  6. The surface
  7. Groupoids
  8. AF-equivalence relations $T^{+}(X_{\mathcal{B}}),T^{+}(Y_{\mathcal{B}} )$
  9. Equivalence relation $T^{\sharp}(Y_{\mathcal{B}})$
  10. Equivalence relation $T^{\sharp}( S^{s}_{\mathcal{B}})$
  11. Equivalence relation $T^{\sharp}( S_{\mathcal{B}})$
  12. The foliation $\mathcal{F}^{+}( S_{\mathcal{B}})$
  13. $C^{*}$-algebras
  14. A Fredholm module
  15. $K$-theory
  16. ...and 12 more sections

Key Result

Lemma 2.7

If $\nu_{r}, \nu_{s}$ is a state on bi-infinite Bratteli diagram, $\mathcal{B}$, then for every integer $n$.

Figures (10)

  • Figure 1: Two presentations of the Chamanara surface: the interiors of the edges with the same label are identified by a translation. The point at the boundary of such edges are not part of the surface and the surface has infinite genus. The presentation on the left is the standard presentation.
  • Figure 2: The functions $\bar{w}_0$ (in black) and $\bar{w}_1$ (in red) on the two presentations of Chamanara's surface from Figure \ref{['fig:Chamanara']}. For $\bar{w}_0$, white is 0, black is 1, and grey is in between, whereas for $\bar{w}_1$ white is 0, red is 1, and the other shades of red are in between.
  • Figure 3: The flat surface defined by the vectors $\zeta_i = (\lambda,\tau)\in \mathbb{R}^\mathcal{A}_+\times T^+_\pi$.
  • Figure 4: The zippered rectangles for the surface in Figure \ref{['fig:1']}.
  • Figure 5: Geometric illustration of Rauzy-Veech induction. Since $\lambda_{\alpha(1)}>\lambda_{\alpha(0)}$, this corresponds to type 1.
  • ...and 5 more figures

Theorems & Definitions (213)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • Proposition 2.9
  • ...and 203 more