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Dominating surface-group representations via Fock-Goncharov coordinates (2405.15378v2)
Published 24 May 2024 in math.GT
Abstract: Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $\rho:\pi_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{C})$, there exists a positive representation $\rho_0:\pi_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\mathbb{X}_n= \mathrm{PSL}_n(\mathbb{C})/\mathrm{PSU}(n)$. Moreover, the $\rho_0$-lengths of peripheral curves remain unchanged. The dominating representation $\rho_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.
- Dylan G. L. Allegretti and Tom Bridgeland. The monodromy of meromorphic projective structures. Trans. Amer. Math. Soc., 373(9):6321–6367, 2020.
- Parameterizing Hitchin components. Duke Math. J., 163(15):2935–2975, 2014.
- Hitchin characters and geodesic laminations. Acta Math., 218(2):201–295, 2017.
- Rajendra Bhatia. Matrix analysis, volume 169 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
- Francesco Brenti. Combinatorics and total positivity. Journal of Combinatorial Theory, Series A, 71(2):175–218, 1995.
- Richard Canary. Anosov representations: Informal lecture notes, 2020. Available at http://www.math.lsa.umich.edu/~canary/Anosovlecnotes.pdf.
- Nonnegative minors of minor matrices. Linear Algebra Appl., 436(7):2187–2200, 2012.
- Moduli spaces of real projective structures on surfaces, volume 38 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2020.
- Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups. Adv. Math., 404:Paper No. 108439, 67, 2022.
- On cyclic Higgs bundles. Math. Ann., 376(3-4):1225–1260, 2020.
- Domination results in n𝑛nitalic_n-Fuchsian fibers in the moduli space of Higgs bundles. Proc. Lond. Math. Soc. (3), 124(4):427–477, 2022.
- Surface groups acting on CAT(−1)CAT1{\rm CAT}(-1)roman_CAT ( - 1 ) spaces. Ergodic Theory Dynam. Systems, 39(7):1843–1856, 2019.
- Daniel C. Douglas. Classical and Quantum Traces Coming from SLn(C) and Uq(sln). ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)–University of Southern California.
- Dominating surface group representations by Fuchsian ones. Int. Math. Res. Not. IMRN, (13):4145–4166, 2016.
- Moduli spaces of local systems and higher teichmüller theory. Publications Mathématiques de l’IHÉS, 103:1–211, 2006.
- Moduli spaces of convex projective structures on surfaces. Adv. Math., 208(1):249–273, 2007.
- Total positivity: tests and parametrizations. Math. Intelligencer, 22(1):23–33, 2000.
- Meromorphic projective structures: Signed spaces, grafting and monodromy. 2023. Preprint. Available at https://arxiv.org/abs/2311.14299.
- Compact anti–de Sitter 3-manifolds and folded hyperbolic structures on surfaces. Pacific J. Math., 275(2):325–359, 2015.
- Spectral networks and snakes. Ann. Henri Poincaré, 15(1):61–141, 2014.
- Dominating surface-group representations into PSL2(ℂ)P𝑆subscript𝐿2ℂ{\mathrm{P}SL}_{2}(\mathbb{C})roman_P italic_S italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_C ) in the relative representation variety. Manuscripta Math., 172(3-4):1169–1186, 2023.
- Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original.
- Nigel Hitchin. Lie groups and Teichmüller space. Topology, 31(3):449–473, 1992.
- Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013.
- François Labourie. Anosov flows, surface groups and curves in projective space. Invent. Math., 165(1):51–114, 2006.
- Bernt Lindström. On the vector representations of induced matroids. Bull. London Math. Soc., 5:85–90, 1973.
- Cross ratios and identities for higher Teichmüller-Thurston theory. Duke Math. J., 149(2):279–345, 2009.
- Carl D. Meyer. Matrix analysis and applied linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, [2023] ©2023.
- d𝑑ditalic_d-pleated surfaces and their shear-bend coordinates. 2023. Available at https://arxiv.org/abs/2305.11780.
- Inequalities: theory of majorization and its applications. Springer Series in Statistics. Springer, New York, second edition, 2011.
- Frederic Palesi. Introduction to positive representations and Fock-Goncharov coordinates. 2013. hal-01218570. Available at https://hal.science/hal-01218570.
- Anne Parreau. Compactification d’espaces de représentations de groupes de type fini. Math. Z., 272(1-2):51–86, 2012.
- Nathaniel Sagman. Infinite energy equivariant harmonic maps, domination, and anti–de Sitter 3-manifolds. J. Differential Geom., 124(3):553–598, 2023.
- Rich Schwartz. The Symmetric Space for SLn(ℝ)𝑆subscript𝐿𝑛ℝSL_{n}(\mathbb{R})italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_R ), 27th November 2013. Math 2710M course notes, available at https://www.math.brown.edu/reschwar/M2710M/symm.pdf.
- Nicolas Tholozan. Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,ℝ)PSL3ℝ{\rm PSL}(3,\mathbb{R})roman_PSL ( 3 , blackboard_R ). Duke Math. J., 166(7):1377–1403, 2017.
- William P. Thurston. Minimal stretch maps between hyperbolic surfaces. In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry, pages 533–585. Amer. Math. Soc., Providence, RI, [2022] ©2022. 1986 preprint, 1998 eprint.
- John von Neumann. A certain zero-sum two-person game equivalent to the optimal assignment problem. In Contributions to the theory of games, vol. 2, volume no. 28 of Ann. of Math. Stud., pages 5–12. Princeton Univ. Press, Princeton, NJ, 1953.
- Anna Wienhard. An invitation to higher Teichmüller theory. In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, pages 1013–1039. World Sci. Publ., Hackensack, NJ, 2018.
- Tengren Zhang. Degeneration of Hitchin representations along internal sequences. 25(5):1588–1645, 2015.
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