相对阿诺索夫群的曼哈顿流形的正则性和精确维数
Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups
浏览论文内容
中文总结 AI 辅助
研究相对阿诺索夫群的帕特森 - 沙利文测度，利用动态方法，证明其相对于特定视觉度量是精确维数的，曼哈顿流形是\(C^1\) - 正则的，增长指标有相应性质，扩展了阿诺索夫表示的相关结论。
中文摘要 AI 辅助
我们建立了关于相对阿诺索夫群的帕特森 - 沙利文测度的几个结果。首先，证明这些测度相对于格罗夫斯 - 曼宁拟等距类中格罗莫夫模型诱导的视觉度量是精确维数的。在群是相对莫尔斯的额外假设下，表明相关的标量卡尔丹度量是格罗莫夫双曲的，相应的边界预度量是精确维数定理适用的视觉度量。其次，证明它们的曼哈顿流形是\(C^1\) - 正则的，由此推断增长指标是\(C^1\) - 正则的且在极限锥内部严格凹。扩展了金 - 吴 - 王关于阿诺索夫表示的情况。我们的方法是动态的，利用了金 - 吴以及布莱亚克 - 卡纳里 - 朱 - 齐默的结果，即相对阿诺索夫群的鲍文 - 马古利斯 - 沙利文测度是有限且混合的。
英文摘要
We establish several results about Patterson--Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves--Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are $C^1$-regular, from which we deduce that the growth indicator is $C^1$-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim--Oh--Wang. Our methods are dynamical, and we exploit the fact due to Kim--Oh and Blayac--Canary--Zhu--Zimmer that Bowen--Margulis--Sullivan measures for relatively Anosov groups are finite and mixing.
