Quotients of surface groups and homology of finite covers via quantum representations

• 成果类型: Article
• 署名作者: Koberda, Thomas; Santharoubane, Ramanujan
• 署名单位: University of Virginia; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0652-x
• 发表日期: 2016
• 页码: 269-292
• 关键词: field theory; integrality; INVARIANTS; tqft

摘要

We prove that for each sufficiently complicated orientable surface S, there exists an infinite image linear representation of such that if is freely homotopic to a simple closed curve on S, then has finite order. Furthermore, we prove that given a sufficiently complicated orientable surface S, there exists a regular finite cover such that is not generated by lifts of simple closed curves on S, and we give a lower bound estimate on the index of the subgroup generated by lifts of simple closed curves. We thus answer two questions posed by Looijenga, and independently by Kent, Kisin, March,, and McMullen. The construction of these representations and covers relies on quantum representations of mapping class groups.