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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
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 ℍ 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 ℍ 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 π 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 = B C k I , 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ℍ 2 并在相关的树中吨 ,证明如下结果。让米 是一个穿孔的双曲环面,让γ 是一个封闭的测地线米 . 由原像中不同测地线形成的任何三角形的任何边γ 在ℍ 2 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π 1 ( 米 ) = 〈 X , 是的 〉 . 我们构造了几个三角形的例子吨 由不同的轴形成吨 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 = 乙 C ķ 一世 , ķ > 0 , 和乙 的终端子词C 和一世 的初始子词C .
更新日期:2019-05-10
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