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Conjugate words and intersections of geodesics in ℍ2
Journal of Topology and Analysis ( IF 0.8 ) Pub Date : 2019-05-10 , DOI: 10.1142/s1793525320500107
Rita Gitik 1
Affiliation  

We investigate intersections of geodesic lines in 2 and in an associated tree T, proving the following result. Let M be a punctured hyperbolic torus and let γ be a closed geodesic in M. Any edge of any triangle formed by distinct geodesic lines in the preimage of γ in 2 is shorter than γ. However, a similar result does not hold in the tree T. Let W be a reduced and cyclically reduced word in π1(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 = BCkI,k > 0, with B a terminal subword of C and I an initial subword of C.

中文翻译:

ℍ2中测地线的共轭词和交点

我们调查测地线的交点2并在相关的树中,证明如下结果。让是一个穿孔的双曲环面,让γ是一个封闭的测地线. 由原像中不同测地线形成的任何三角形的任何边γ2短于γ. 然而,类似的结果在树 T 中并不成立。让W是一个减少和循环减少的词π1() = X,是的. 我们构造了几个三角形的例子由不同的轴形成通过共轭稳定W使得这些三角形中的一条边长于大号(W). 我们还证明如果W重叠它的两个共轭,使得重叠覆盖所有W并且重叠不相交,则存在分解W = Cķ一世,ķ > 0, 和的终端子词C一世的初始子词C.
更新日期:2019-05-10
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