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Period collapse of Markov triangles

Marc Fares

TL;DR

This work generalizes the complete period collapse phenomenon from Fibonacci triangles to all Markov triangles by leveraging integral affine geometry and geometry-of-mutions. It analyzes Ehrhart quasipolynomials of Markov triangles through geometric mutations, integral barycentres, and standard-position normalization, showing that rational cases have period dividing the largest Markov-related index and that limiting irrational cases are pseudo-rational with a prescribed period. The authors construct two-sided mutation sequences for each Markov number $p$ yielding two rational triangles and two irrational limits whose Ehrhart function has exact period $p$, demonstrating strong period collapse in both finite and limiting regimes. They further connect these combinatorial-geometric phenomena to symplectic geometry, where these triangles model bases of almost toric fibrations in $\mathbb{CP}^2$ and relate to symplectic embedding staircases, underscoring deep links between Markov numbers, Ehrhart theory, and symplectic topology. Overall, the paper broadens the scope of period-collapse phenomena and reveals new structural ties between discrete geometry and symplectic embedding problems.

Abstract

Cristofaro-Gardiner and Kleinman showed the complete period collapse of the Ehrhart quasipolynomial of Fibonacci triangles and their irrational limits, by studying the Fourier-Dedekind sums involved in the Ehrhart function of right-angled rational triangles. We generalize this result using integral affine geometrical methods to all Markov triangles, as defined by Vianna. In particular, we show new occurrences of strong period collapse, namely by constructing for each Markov number $p$ a two-sided sequence of rational triangles and two irrational limits with quasipolynomial Ehrhart function of period $p$.

Period collapse of Markov triangles

TL;DR

This work generalizes the complete period collapse phenomenon from Fibonacci triangles to all Markov triangles by leveraging integral affine geometry and geometry-of-mutions. It analyzes Ehrhart quasipolynomials of Markov triangles through geometric mutations, integral barycentres, and standard-position normalization, showing that rational cases have period dividing the largest Markov-related index and that limiting irrational cases are pseudo-rational with a prescribed period. The authors construct two-sided mutation sequences for each Markov number yielding two rational triangles and two irrational limits whose Ehrhart function has exact period , demonstrating strong period collapse in both finite and limiting regimes. They further connect these combinatorial-geometric phenomena to symplectic geometry, where these triangles model bases of almost toric fibrations in and relate to symplectic embedding staircases, underscoring deep links between Markov numbers, Ehrhart theory, and symplectic topology. Overall, the paper broadens the scope of period-collapse phenomena and reveals new structural ties between discrete geometry and symplectic embedding problems.

Abstract

Cristofaro-Gardiner and Kleinman showed the complete period collapse of the Ehrhart quasipolynomial of Fibonacci triangles and their irrational limits, by studying the Fourier-Dedekind sums involved in the Ehrhart function of right-angled rational triangles. We generalize this result using integral affine geometrical methods to all Markov triangles, as defined by Vianna. In particular, we show new occurrences of strong period collapse, namely by constructing for each Markov number a two-sided sequence of rational triangles and two irrational limits with quasipolynomial Ehrhart function of period .
Paper Structure (22 sections, 19 theorems, 44 equations, 9 figures)

This paper contains 22 sections, 19 theorems, 44 equations, 9 figures.

Table of Contents

  1. Introduction and main results
  2. Outline of the paper.
  3. Integral affine geometry
  4. Basic concepts.
  5. Half-shears.
  6. Ehrhart theory
  7. Basic notions.
  8. Period collapse.
  9. Markov triangles
  10. The Markov tree.
  11. Markov triangles.
  12. Ehrhart theory of Markov triangles
  13. The rational case.
  14. The limiting case.
  15. Open problem.
  16. ...and 7 more sections

Key Result

Proposition 1.5

Let $v$ be an integral vector, and consider the half-shear $\varphi^{1/2}_v$ along $v$. Then $\varphi^{1/2}_v(P)$ and $P$ are Ehrhart equivalent.

Figures (9)

  • Figure 1: Integrally congruent triangles.
  • Figure 2: Example of a triangle with denominator $24$
  • Figure 3: The limiting Fibonacci triangle.
  • Figure 4: The first five generations of the Markov tree.
  • Figure 5: Geometric mutation of a $(1,5,2)$-triangle at the $1$-vertex.
  • ...and 4 more figures

Theorems & Definitions (47)

  • definition 1.2
  • remark 1.3
  • Proposition 1.5
  • Theorem 1.6: Ehrhart
  • definition 1.7
  • Theorem 1.8
  • remark 1.12
  • definition 1.14
  • remark 1.17
  • remark 1.18
  • ...and 37 more