Bounded geodesic image theorem via bicorn curves

Thurston 测地线：无回溯与活跃区间

Thurston geodesics: no backtracking and active intervals

本文在 Thurston 度量下定义子曲面沿测地线的活跃区间，证明其间短曲线形成无回溯拟测地线、区间外曲线图移动一致有界。

紧测地线与稳定长度的组合学

Combinatorics of tight geodesics and stable lengths

给出了一个计算伪Anosovs在曲线图上的稳定长度的算法，回答了Bowditch的一个问题。给出了计算伪Anosovs的所有不变紧测地轴的方法。一路上，我们表明，有常数$1<a_1<a_2$，这样的最小上界紧测地线的“片”是有界的下面和上面的$a_1^{ξ(S)}$和$a_2^{ξ(S)}$，其中$ξ(S)$是表面的复杂性。作为结果，我们给出了曲线图和映射类群的渐近维数的第一可计算界。我们的技术涉及一个概括的Masur-明斯基的紧测地线和一类新的路径上，他们的收紧程序的作品。

超稠密性与模空间中的有界测地线

Superdensity and bounded geodesics in moduli space

本文定义流的超稠密性，证明平移曲面上线性流超稠密当且仅当其对应 Teichmüller 测地线上曲面直径有界，从而推广 Beck-Chen 的结果。

上有界曲率空间中的闭测地线

Closed geodesics in Alexandrov spaces of curvature bounded from above

本文证明了$W^{1,2}$映射到$CAT(K)$空间的局部能量凸性。这种能量凸性使我们能够扩展Colding和Minicozzi的宽度扫掠构造，以在任何从上方有界的曲率封闭亚历山德罗夫空间中产生闭测地线，这也提供了关于亚历山德罗夫设置中存在非平凡闭测地线的Birkhoff-Lyusternik定理的广义版本。

Weak tight geodesics in the curve complex: examples and gaps

Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics between any two vertices in the curve complex. Bowditch found a uniform finiteness property on tight geodesics [9], and this property has given further important applications in some of the above studies. In this paper, we introduce classes of geodesics which are not tight but still …

Dehn钻井作业下的体积增加界限

Bounds on Volume Increase under Dehn Drilling Operations

本文研究了双曲流形在去除一条曲线即Dehn钻孔的过程中体积是如何增加的。如果我们删除的曲线是测地线，我们能够表明，对于某个家庭的流形，体积的增加是有界的$π\cdot l$以上，其中$l$是长度的测地线钻。此外，我们构造的例子表明，有没有下界的体积增加的长度的正幂的线性函数，特别是体积增加的长度不是线性有界的。

裤子图上的稳定线性方程是有理的

Stable Lengths on the pants graph are rational

对于裤子图，很少有人知道测地线的行为，而不是准测地线。Brock-Masur-Minsky证明了连接端点的测地线或测线段满足有界组合条件，例如伪Anosov的稳定/不稳定叠层，都有界组合，\textit{outside of annuli}。在本文中，它是存在的测地线，也有界组合环内。这些测地线被证明具有有限性质类似于那些紧测地线在复杂的曲线，从合理性的稳定长度的伪Anosovs的裤子图，然后遵循从参数的Bowditch的曲线复杂。

有界几何的带边界流形

Manifolds with boundary and of bounded geometry

对于有边界的非紧流形，我们证明了由坐标自由曲率界定义的有界几何等价于由测地坐标中度量张量的界定义的有界几何.我们在有界几何边界的流形上构造了一个具有从属单位单位分解的地图集，并研究了测地坐标地图的变化。

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic set…

$L^2$-界用于在凸余紧双曲3-流形中钻孔短测地线

$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds

本文研究了凸余紧双曲3-流形钻孔短测地线后边界复射影结构的变化，给出了不依赖于边界单射半径的$L^2$-界，证明其变化受普适常数乘以测地线长度平方根的约束。

上界Pants分解上的Fenchel-Nielsen坐标

Fenchel-Nielsen coordinates on upper bounded pants decompositions

设$X_0$是一个具有上界Pants分解的无限亏格双曲曲面（其边界分量是闭测地线或穿孔）.长度谱Teichmüller空间$T_{ls}(X_0)$由所有与$X_0$同胚的曲面$X$组成，使得相应的简单闭测地线的比率从下到上一致有界。Alessandrini，Liu，Papadopoulos和Su描述了$T_{ls}(X_0)$的Fenchel-Nielsen坐标，并使用这些坐标证明了$T_{ls}(X_0)$是路径连通的。我们使用$T_{ls}(X_0)$的Fenchel-Nielsen坐标来诱导$l^{\infty}$和$T_{ls}(X_0)$之间的局部双Lipschitz同胚（这推广了Fletcher和Allessandrini，Liu，Papadopoulos，Su和Sun对于未约化和约化的$T_{qc}(X_0)$的类似结果）。因此，$T_{ls}(X_0)$是可收缩的。我们还刻画了拟共形Teichmüller空间$T_{qc}(X_0)$在$T_{ls}(X_0)$中的长度谱度量中的闭包.