共形维数界、庞特里亚金球面边界与直角考克斯特群的代数纤维化
Conformal dimension bounds, Pontryagin sphere boundaries, and algebraic fibering of right-angled Coxeter groups
AI总结：
研究直角考克斯特群，通过引入\((n,m)\) - 分支条件构造组合圆树并嵌入戴维斯复形，给出双曲群边界共形维数下界，展示相关图族，证明嵌入结果，得出群的共形维数等结论，升级相关工作。
摘要：
我们引入一种称为\((n,m)\) - 分支的图论条件，它确保具有受控分支参数的组合圆树能拟等距嵌入由该图定义的直角考克斯特群的戴维斯复形中。此构造给出了此类双曲群边界共形维数的下界。我们展示了许多具有此性质的图族，包括许多一维球面建筑。我们证明了一个嵌入结果，表明在温和假设下，无平方旗图可嵌入闭曲面的无平方旗三角剖分的诱导子图中。我们用此将分支图嵌入到表示具有庞特里亚金球面边界的双曲直角考克斯特群的图中。我们得出存在共形维数趋于无穷的此类群的例子，因此该族中有无限多个拟等距类。我们用共形维数表明，Lafont - Minemyer - Sorcar - Stover - Wells的近期工作可升级以得出，对于每个\(n\geq2\)，存在无限多个虚拟代数纤维化且虚拟上同调维数为\(n\)的双曲直角考克斯特群的拟等距类。
英文摘要：
We introduce a graph-theoretic condition, called $(n,m)$--branching, that ensures a combinatorial round tree with controlled branching parameters can be quasi-isometrically embedded in the Davis complex of the right-angled Coxeter group defined by the graph. This construction yields a lower bound on the conformal dimension of the boundary of such a hyperbolic group. We exhibit numerous families of graphs with this property, including many 1-dimensional spherical buildings. We prove an embedding result, showing that under mild hypotheses a flag-no-square graph embeds as an induced subgraph in a flag-no-square triangulation of a closed surface. We use this to embed our branching graphs into graphs presenting hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary. We conclude there are examples of such groups with conformal dimension tending to infinity, and hence, there are infinitely many quasi-isometry classes within this family. We use conformal dimension to show that recent work of Lafont--Minemyer--Sorcar--Stover--Wells can be upgraded to conclude that for every $n \geq 2$ there exist infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups that virtually algebraically fiber and have virtual cohomological dimension $n$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
