苏州双曲几何研讨会---学术报告(2024.10.12~13)
会议报告地点：苏州大学天赐庄校区本部维格堂319
2024.10.12
09:30-10:20 伍晓磊：Embedding groups into simple groups
10:50-11:40 郑芳婷：Recent progress on high-dimensional hyperbolic reflection groups
2024.10.13
08:50-09:40 陈史标：Punctured torus groups, generalised Gauss maps and billiards on hyperbolic quadrilaterals
10:00-10:50马继明：The Menger curve and spherical CR-uniformization of a closed hyperbolic 3-orbifold
11:10-12:00 孙 哲：Exponential volumes of moduli spaces of hyperbolic surfaces
报告摘要
报告题目：Embedding groups into simple groups
报告人：伍晓磊（复旦大学）
摘要： Simple groups have played an important role in the studying of groups. The Boone-Higman Conjecture says a finitely generated group can be embedded into a finitely presented simple group if and only if it has solvable word problem. The conjecture has now been proved for many classes of groups, including hyperbolic groups, RAAGs, linear groups over rationals.
In this talk, we will discuss a strategy that can be used to embed groups acting faithfully on locally finite trees to finitely presented simple groups. This in particular verifies the conjecture for the Baumslag-Solitar groups and free-by-cyclic groups. This is a joint work with Kai-Uwe Bux and Claudio Llosa Isenrich.
报告题目：Recent progress on high-dimensional hyperbolic reflection groups
报告人：郑芳婷（西交利物浦大学）
摘要：In this talk, I will provide a brief overview of hyperbolic reflection groups, and conclude with some new results that have been achieved in recent years, particularly in high dimensions. Our contributions will be discussed in the end, which are based on a series of joint works with Jiming Ma.
报告题目：Punctured torus groups, generalised Gauss maps and billiards on hyperbolic quadrilaterals
报告人：陈史标（新加坡国立大学）
摘要：We consider certain families of objects parametrization by rationals in (0,1) from different viewpoints: geometric, group theoretic and dynamical and explore the orbit decomposition of the rationals for different parameters in the parameter space. In particular, we show how various questions like the pseudomodularity of punctured torus groups, equivalence classes of generalised Gauss maps on the unit interval and billiards on ideal hyperbolic quadrilaterals are related. This is joint work with Nhat Minh Doan and Wei Daren.
报告题目：The Menger curve and spherical CR-uniformization of a closed hyperbolic 3-orbifold
报告人：马继明（复旦大学）
摘要：Let G_{6,3} be a hyperbolic polygon-group with boundary the Menger curve. Granier in 2015 constructed a discrete, convex cocompact and faithful representation ρ of G_{6,3}
into PU(2,1). We show the 3-orbifold at infinity of ρ(G_{6,3}) is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the Z_3-coned chain-link C(6,-2). This answers the second part of a Kapovich's Conjecture in 2022, and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure after Schwartz's first example in 2003. This is joint work with Baohua Xie.
报告题目：Exponential volumes of moduli spaces of hyperbolic surfaces
报告人：孙哲（中国科学技术大学）
摘要：Mirzakhani found a remarkable recursive formula for the volumes of the moduli spaces of the hyperbolic surfaces with geodesic boundary, and the recursive formula plays very important role in several areas of mathematics: topological recursion, random hyperbolic surfaces etc. We consider some more general moduli spaces M_S(K,L) where the hyperbolic surfaces would have crown ends and horocycle decorations at each ideal points. But the volume of the space M_S(K,L) is infinite when S has the crown ends. To fix this problem, we introduce the exponential volume form given by the volume form multiplied by the exponent of a canonical function on M_S(K,L). We show that the exponential volume is finite. And we prove the recursion formulas for the exponential volumes, generalising Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We expect the exponential volumes are relevant to the open string theory. This is a joint work with Alexander Goncharov.
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