Classification of trees that quasi-inscribe rectangles in the hyperbolic plane

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 ℝ 2 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 2 π − 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 .