论文标题
$ \ mathrm {so} _0（2，n+1）$ - 最大表示和双曲表面
$\mathrm{SO}_0(2,n+1)$-maximal representations and hyperbolic surfaces
论文作者
论文摘要
我们研究表面组的最大表示$ρ：π_1（σ）\ to \ mathrm {so} _0（2，n+1）$，通过引入$ρ$ invariant psseudo-riemannian Space $ \ mathbb $ \ mathbb {h}^2，n} $ geod的$ pseudo-riemannian Space $ \ mathbb {h} $ geod for pseudo-riemannian space $ \ mathbb n} $ max lam geod。 我们证明，$ρ$ - 易变的百褶表面总是嵌入，可aus骨，并且具有内在的伪金属和双曲线结构。我们通过构建从自然相关的交叉比率到$ρ$的剪切合过程来描述后者。为此目的而开发的过程适用于广泛的交叉比例，包括$ \ m atrm {so}（p，q）$中的Hitchin和$θ$阳性表示形式。我们还表明，$ρ$的长度光谱占据了$ρ$ invariant褶皱的表面，严格的不平等恰好在与弯曲基因座相交的曲线上。 我们观察到，$ρ$ - invariant的百褶表面的规范分解与叶子和斑块对应于guichard-wienhard域的分解$ρ$的guichard-wienhard域中，分为标准的纤维块中，即三角形和光子线。相反，我们通过将光子的三角形将光子表面的光子歧管构造在光纤表面上。 我们开发的工具允许Collier，Tholozan和Toulisse在$ρ$的（伪里人）几何形状上恢复各种结果，并通过建设性和几何形状方法与最大代表和纤维光子歧管之间的对应关系，绕过Higgs Bundles的使用。
We study maximal representations of surface groups $ρ:π_1(Σ)\to\mathrm{SO}_0(2,n+1)$ via the introduction of $ρ$-invariant pleated surfaces inside the pseudo-Riemannian space $\mathbb{H}^{2,n}$ associated to maximal geodesic laminations of $Σ$. We prove that $ρ$-invariant pleated surfaces are always embedded, acausal, and possess an intrinsic pseudo-metric and a hyperbolic structure. We describe the latter by constructing a shear cocycle from the cross ratio naturally associated to $ρ$. The process developed to this purpose applies to a wide class of cross ratios, including examples arising from Hitchin and $Θ$-positive representations in $\mathrm{SO}(p,q)$. We also show that the length spectrum of $ρ$ dominates the ones of $ρ$-invariant pleated surfaces, with strict inequality exactly on curves that intersect the bending locus. We observe that the canonical decomposition of a $ρ$-invariant pleated surface into leaves and plaques corresponds to a decomposition of the Guichard-Wienhard domain of discontinuity of $ρ$ into standard fibered blocks, namely triangles and lines of photons. Conversely, we give a concrete construction of photon manifolds fibering over hyperbolic surfaces by gluing together triangles of photons. The tools we develop allow to recover various results by Collier, Tholozan, and Toulisse on the (pseudo-Riemannian) geometry of $ρ$ and on the correspondence between maximal representations and fibered photon manifolds through a constructive and geometric approach, bypassing the use of Higgs bundles.
