标题： 某些柯克伍德群博德奇边界的共形维数界限 显示英文标题

标题： Conformal dimension bounds for certain Coxeter group Bowditch boundaries

摘要： 我们给出了定义图是完全图且边标签至少为三的Coxeter群的Bowditch边界共形维数的上下界。 下界是通过在Davis复形中拟等距嵌入Gromov的圆树得到的。 上界是通过在CAT(-1)空间上展示一个几何有限作用并限制该空间视觉边界的Hausdorff维数来给出的。 我们的结果表明，在每个边标签有上界的此类Coxeter群的无限族中，存在无限多个拟等距类。 作为应用，我们证明在具有Pontryagin球面边界的双曲群族中，存在无限多个拟等距类。 将我们的结果与Bourdon--Kleiner的工作结合，证明了该族中双曲群边界的共形维数在$(1,\infty)$中达到一个稠密集。 显示英文摘要

摘要： We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding Gromov's round trees in the Davis complex. The upper bounds are given by exhibiting a geometrically finite action on a CAT(-1) space and bounding the Hausdorff dimension of the visual boundary of this space. Our results imply that there are infinitely many quasi-isometry classes within each infinite family of such Coxeter groups with edge labels bounded from above. As an application, we prove there are infinitely many quasi-isometry classes among the family of hyperbolic groups with Pontryagin sphere boundary. Combining our results with work of Bourdon--Kleiner proves the conformal dimension of the boundaries of hyperbolic groups in this family achieves a dense set in $(1,\infty)$.