学术报告报告题目：Schwartz's complex hyperbolic surface
报 告人：马继明副教授（复旦大学）
报告时间：2022年11月22日（星期二）上午9:30-11:00
报告地点：zoom 98951767939，密码：GTG
邀 请人：杜晓明副教授
欢迎广大师生前往！
数学学院
2022年11 月17日
报告摘要： In 2003 R. Schwartz consider an arithmetic geometrically finite, discrete subgroup of $PU(2,1)$. This subgroup is the image of a representation $\rho: G(4,7) \to PU(2,1)$. He also determined the 3-manifold at infinity of the quotient $H^2_C/\rho(G(4,7))$. It is a closed hyperbolic 3-orbifold with underlying space the 3-sphere and whose singularity locus is a two-components link equipped with a $Z_2$-cone structure. Even until today, this orbifold is the unique explicit closed hyperbolic 3-orbifold admitting uniformizable spherical CR-structures.
In this talk, we show the representation $\rho: G(4,7)\to PU(2,1)$ is faithful. More importantly, we determine the 4-dimensional topology of the complex hyperbolic surface $H^2_C/\rho(G(4,7))$ via a symmetric, beautiful and simplest handle structure. More precisely, let $\Sigma(4,7)$ be the index two even subgroup of $\rho(G(4,7))$, then $H^2_C/\Sigma(4,7)$ is a 2-dimensional complex hyperbolic orbifold with six isolated singularities, small closed neighborhoods of these singularities are cones on lens spaces $L(4,-1)$ and $L(7,-1)$ respectively. We prove that $H^2_C/\Sigma(4,7)$ can be obtained from the small neighborhoods of the six isolated singularities by attaching nine 1-handles and eight 2-handles.