Geometry and topology of curves and surfaces in closed hyperbolic manifolds

PI将研究闭双曲流形的几何和拓扑问题。结合虚哈肯猜想，我们将讨论双曲三维流形是否包含大量具有不可压缩边界的等分布本质三维流形的问题。本文研究高维双曲流形，目的是证明每一个至少4维的闭双曲流形都包含一个本质3-流形群。此外，它将表明，简单回路猜想失败的每一个维度大于或等于4。相关的问题（特别是曲面子群猜想）将在其他双曲空间中得到解决，如复双曲空间或双曲群。纯数学是后来用于物理学和生物学等自然科学的思想的温床。在物理学中，宇宙被描述为三维空间，因此研究三维流形的几何和拓扑在我们所面临的真实的生活物理问题中可能是非常重要的。

The PI will study questions about geometry and topology of closed hyperbolic manifolds. In connection with the Virtual Haken Conjecture, the question of whether a hyperbolic 3-manifold contains an abundance of equidistributed essential 3-manifolds with incompressible boundary will be addressed. Higher dimensional hyperbolic manifolds will be studied and the aim is to prove that every closed hyperbolic manifold of dimension at least 4 contains an essential 3-manifold group. Also, it will be shown that the Simple Loop Conjecture fails in every dimension greater than or equal to 4. Related problems (in particular the Surface Subgroup Conjecture) will be address for other hyperbolic spaces, like complex hyperbolic spaces or hyperbolic groups.Pure mathematics is a breeding ground for ideas that are later utilized in natural sciences like physics and biology. In physics, the universe is described as a 3-dimensional space, thus studying geometry and topology of 3-manifolds may prove very important in the real life physical questions that we face.