Volume 26, issue 2 (2022)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 29, 1 issue Volume 29, 1 issue

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Surface group representations in ${\rm SL}_2(\mathbb{C})$ with finite mapping class orbits

Indranil Biswas, Subhojoy Gupta, Mahan Mj and Junho Peter Whang

Geometry & Topology 26 (2022) 679–719
Bibliography
1 I Biswas, T Koberda, M Mj, R Santharoubane, Representations of surface groups with finite mapping class group orbits, New York J. Math. 24 (2018) 241 MR3778503
2 P Boalch, The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math. 596 (2006) 183 MR2254812
3 J Bourgain, A Gamburd, P Sarnak, Markoff triples and strong approximation, C. R. Math. Acad. Sci. Paris 354 (2016) 131 MR3456887
4 S Cantat, F Loray, Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation, Ann. Inst. Fourier (Grenoble) 59 (2009) 2927 MR2649343
5 D Cooper, J F Manning, Nonfaithful representations of surface groups into SL(2, ) which kill no simple closed curve, Geom. Dedicata 177 (2015) 165 MR3370029
6 G Cousin, Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties, Math. Ann. 367 (2017) 965 MR3623216
7 G Cousin, V Heu, Algebraic isomonodromic deformations and the mapping class group, J. Inst. Math. Jussieu 20 (2021) 1497 MR4311561
8 S K Donaldson, Twisted harmonic maps and the self-duality equations, Proc. Lond. Math. Soc. 55 (1987) 127 MR887285
9 B Dubrovin, M Mazzocco, Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000) 55 MR1767271
10 R Fuchs, Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann. 63 (1907) 301 MR1511408
11 D Gallo, M Kapovich, A Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. 151 (2000) 625 MR1765706
12 W M Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, from: "Handbook of Teichmüller theory, II" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc. (2009) 611 MR2497777
13 N J Hitchin, Poncelet polygons and the Painlevé equations, from: "Geometry and analysis" (editor S S Abhyankar), Tata Inst. Fund. Res. (1995) 151 MR1351506
14 R D Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972) 635 MR314993
15 K Iwasaki, A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002) 131 MR1930217
16 T Koberda, R Santharoubane, Quotients of surface groups and homology of finite covers via quantum representations, Invent. Math. 206 (2016) 269 MR3570293
17 M Laurent, Équations diophantiennes exponentielles, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 945 MR777582
18 O Lisovyy, Y Tykhyy, Algebraic solutions of the sixth Painlevé equation, J. Geom. Phys. 85 (2014) 124 MR3253555
19 J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures, I, Ann. of Math. 120 (1984) 401 MR769158
20 A Patel, A N Shankar, J P Whang, The rank two p–curvature conjecture on generic curves, Adv. Math. 386 (2021) MR4270046
21 J P Previte, E Z Xia, Topological dynamics on moduli spaces, II, Trans. Amer. Math. Soc. 354 (2002) 2475 MR1885660
22 J H Przytycki, A S Sikora, On skein algebras and Sl2()–character varieties, Topology 39 (2000) 115 MR1710996
23 M van der Put, M F Singer, Galois theory of linear differential equations, 328, Springer (2003) MR1960772
24 K Saito, Character variety of representations of a finitely generated group in SL2, from: "Topology and Teichmüller spaces" (editors S Kojima, Y Matsumoto, K Saito, M Seppälä), World Sci. (1996) 253 MR1659663
25 C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, II, Inst. Hautes Études Sci. Publ. Math. 80 (1994) 5 MR1320603
26 J C Sinz, Some results in representation dynamics, PhD thesis, University of Chicago (2010) MR2821997
27 S P Tan, Y L Wong, Y Zhang, End invariants for SL(2, ) characters of the one-holed torus, Amer. J. Math. 130 (2008) 385 MR2405161
28 H Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Sup. 6 (1889) 3 MR1508833
29 J P Whang, Diophantine analysis on moduli of local systems, PhD thesis, Princeton University (2018) MR3828518
30 J P Whang, Global geometry on moduli of local systems for surfaces with boundary, Compos. Math. 156 (2020) 1517 MR4157427
31 J P Whang, Nonlinear descent on moduli of local systems, Israel J. Math. 240 (2020) 935 MR4193155