Conway's topographic approach to binary quadratic forms and Markov triples is reviewed
from the point of view of the theory of two-valued groups. This leads naturally to a new class …

We study the growth of the values of integer binary quadratic forms Q on a binary planar tree
as it was described by Conway. We show that the corresponding Lyapunov exponents Λ Q …

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to
the modular group. Let be a finite index subgroup of with an action on a complex simple Lie …

A celebrated result concerning triangulations of a given closed three-manifold is that any two
triangulations with the same number of vertices are connected by a sequence of so-called 2 …

On a hyperbolic surface homeomorphic to a torus with a puncture, each oriented simple
geodesic inherits a well-defined relative twist number in $[0, 1] $, given by the ratio to its …

Изучение значений бинарных квадратичных форм–классическая тема, восходящая к
Лежандру и Гауссу. Вопрос о числах N, которые можно представить в виде суммы …

We discuss the relationship between Penner's $\lambda $-length and the norms of
Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that …

We classify and enumerate all rational numbers with approximation constant at least 1 3
using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus …

Combining McShane's identity on a hyperbolic punctured torus with well-known geometric
interpretations of the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent …

This paper describes a tree structure on the set of orthogeodesics leading to a combinatorial
proof of Basmajian's identity in the case of surfaces. It is motivated by a number theoretic …