Conjugate Words and Intersections of Geodesics in $H^2$

TL;DR

This paper explores intersections of geodesic lines in hyperbolic space and associated trees, revealing differences in geometric properties and providing new constructions and decompositions related to words in fundamental groups.

Contribution

It establishes a new geometric inequality for geodesics in hyperbolic surfaces and constructs counterexamples in trees, along with a novel word decomposition result.

Findings

Edges in triangles formed by geodesics are shorter than the closed geodesic

Counterexamples show edges can be longer in associated trees

A specific word decomposition based on overlaps is proven

Abstract

We investigate intersections of geodesic lines in $H^2$ and in an associated tree T, proving the following result. Let M be a punctured hyperbolic torus and let $\gamma$ be a closed geodesic in M. Any edge of any triangle formed by distinct geodesic lines in the preimage of $\gamma$ in $H^2$ is shorter then $\gamma$. However, a similar result does not hold in the tree T. Let W be a reduced and cyclically reduced word in $\pi_1(M) = <x,y>$. We construct several examples of triangles in T formed by distinct axes in T stabilized by conjugates of W such that an edge in those triangles is longer than L(W). We also prove that if W overlaps two of its conjugates in such a way that the overlaps cover all of W and the overlaps do not intersect, then there exists a decomposition $W = BC^kI; k > 0$, with B a terminal subword of C and I an initial subword of C.