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

• Published in: Inventiones mathematicae Vol. 206; no. 2; pp. 269 - 292
• Main Authors: Koberda, Thomas, Santharoubane, Ramanujan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-11-2016
• Subjects: Estimates; Lifts; Lower bounds; Mathematical analysis; Mathematics; Mathematics and Statistics; Representations; Subgroups; Texts; Surface Group; Mapping Class Group; Finite Order; Boundary Component; Primitive Root
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-016-0652-x

We prove that for each sufficiently complicated orientable surface
S
, there exists an infinite image linear representation
ρ
of
π
1
(
S
)
such that if
γ
∈
π
1
(
S
)
is freely homotopic to a simple closed curve on
S
, then
ρ
(
γ
)
has finite order. Furthermore, we prove that given a sufficiently complicated orientable surface
S
, there exists a regular finite cover
S
′
→
S
such that
H
1
(
S
′
,
Z
)
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, Marché, and McMullen. The construction of these representations and covers relies on quantum
SO
(
3
)
representations of mapping class groups.