Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
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Juan Pablo Díaz
und
Rogelio Valdez
Abstract
The inscribed square problem asks if every Jordan curve contains the four vertices of a non-degenerate Euclidean square. This question was posed by Otto Toeplitz in 1911 and remains open to this date. It is known that, with the exception of the arc and the 3-star, every topological copy of a locally connected plane continuum in ℝ2 contains the four vertices of at least one Euclidean rectangle; see [13]. Given ε > 0, a hyperbolic ε-rectangle is a hyperbolic quadrilateral such that its inner angles sum up to more than 2π − ε, and such that its diagonals share the midpoint and have the same hyperbolic length. A tree T quasi-inscribes rectangles if for every ε > 0, any embedding of T in the hyperbolic plane admits an inscribed ε-rectangle. In this paper, we classify trees that quasi-inscribe rectangles. We also show that any Jordan curve J in the hyperbolic plane admits at least one equilateral hyperbolic triangle whose area is less than π − K, where K is a positive constant that depends only on the diameter of J.
Acknowledgements
The authors thank the referee for the very careful reading of the manuscript and the valuable suggestions made that improved the paper.
Communicated by: R. Löwen
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
