论文标题
痕量收缩和下沉常数
Trace Systoles and Sink Constant
论文作者
论文摘要
令$σ$为$χ（σ）<0 $的表面，而来自基本组$π_1（σ）$的表示$ρ$ to $ \ rm {sl}（2，\ mathbb {c}）$。我们定义$ρ$的\ emph {trace systole}，表示为$ \ mathrm {tys}（ρ）$作为folows：$ \ mathrm {tys}（tys}（ρ）= \ inf \ inf \ left \ left \ left \ {| \ rm {tr}（ρ（γ））| \ \，\γ\在π_1（s）\ mbox {必需简单的封闭曲线} \ right \} $$当$σ$赋予双曲线结构时，拟态表示的痕量收缩期是与该研究动力的一部分相关性相关的。函数$ \ mathrm {tys} $在$σ$的相对角色品种上有限，在本文中，我们明确地计算了一个单孔的圆环，四孔的球体和不可定向的表面$ 3 $ 3 $ 3 $。证明依赖于鲍迪奇（Bowditch）引入的这些表面组的表示与所谓的Markoff图之间的对应关系。由此，我们推断出对某些双曲线歧管的最佳收缩期不平等以及这些表面的非瞬间表示。
Let $Σ$ be a surface with $χ(Σ) < 0$, and a representation $ρ$ from the fundamental group $π_1 (Σ)$ into $ \rm{SL} (2 , \mathbb{C})$. We define the \emph{trace systole} of $ρ$, denoted $\mathrm{tys} (ρ)$ as folows : $$\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.
