迹收缩长度与汇常数
Trace Systoles and Sink Constant
AI总结：
该研究定义了曲面基本群到SL(2,ℂ)表示的迹收缩长度，借助Markoff映射对应关系，显式算出三类曲面相对特征簇上的最优界，并得到双曲流形收缩不等式与非Fuchsian表示的相关推论。
摘要：
设Σ是满足χ(Σ)<0的曲面，ρ是从基本群π₁(Σ)到SL(2,ℂ)的表示。我们定义ρ的迹收缩长度（trace systole），记为tys(ρ)，定义如下：$$\mathrm{tys}(\rho) = \inf \left\{ | \mathrm{tr}(\rho(\gamma)) | \\ , \\ \gamma\in \pi_1(S) \mbox{ 本质简单闭曲线} \right\}$$当Σ配备双曲结构时，完整和乐表示的迹收缩长度与双曲曲面通常的收缩长度自然相关，这是本研究的动机之一。tys函数在Σ的相对特征簇上有界，本文中我们显式计算了单洞环面、四洞球面和亏格3的不可定向曲面的最优界。证明依赖于这些曲面群的表示与Bowditch提出的所谓Markoff映射之间的对应关系。由此，我们推导出关于某些双曲流形的最优收缩不等式，以及这些曲面的非Fuchsian表示的各种推论。
英文摘要：
Let $Σ$ be a surface with $χ(Σ) < 0$, and a representation $ρ$ from the fundamental group $π_1 (Σ)$ into $ \rm{SL} (2 , \mathbb{C})$. We define the \emph{trace systole} of $ρ$, denoted $\mathrm{tys} (ρ)$ as folows : $$\mathrm{tys} (ρ) = \inf \left\{ | \rm{tr} (ρ(γ)) | \ , \ γ\in π_1 (S) \mbox{ essential simple closed curve} \right\}$$ When $Σ$ is endowed with an hyperbolic structure, the trace systole of the holonomy representation is naturally related to the usual systolic length of the hyperbolic surface, which is one of the motivation for this study. The function $\mathrm{tys}$ is bounded on relative character varieties of $Σ$, and in this article we compute explicitly the optimal bounds for the one-holed torus, the four-holed sphere and the non-orientable surface of genus $3$. The proofs rely on the correspondance between representations of these surface groups and so-called Markoff maps which were introduced by Bowditch. From this, we infer various consequences on the optimal systolic inequalities of certain hyperbolic manifolds and also on non-Fuchsian representations for these surfaces.
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。
