## Geometry & Topology

### On type-preserving representations of the four-punctured sphere group

Tian Yang

#### Abstract

We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation $ρ: π1(Σg,n) → PSL(2; ℝ)$ of a punctured surface group that sends every nonperipheral simple closed curve to a hyperbolic element must $ρ$ be Fuchsian. The counterexamples come from relative Euler class $± 1$ representations of the four-punctured sphere group. We also show that the mapping class group action on each nonextremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic, which confirms a conjecture of Goldman for this case. The main tool we use are Kashaev and Penner’s lengths coordinates of the decorated character spaces.

#### Article information

Source
Geom. Topol., Volume 20, Number 2 (2016), 1213-1255.

Dates
Revised: 14 May 2015
Accepted: 4 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858977

Digital Object Identifier
doi:10.2140/gt.2016.20.1213

Mathematical Reviews number (MathSciNet)
MR3493103

Zentralblatt MATH identifier
1347.57002

#### Citation

Yang, Tian. On type-preserving representations of the four-punctured sphere group. Geom. Topol. 20 (2016), no. 2, 1213--1255. doi:10.2140/gt.2016.20.1213. https://projecteuclid.org/euclid.gt/1510858977

#### References

• F Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. 5 (1996) 233–297
• F Bonahon, Low-dimensional geometry: From Euclidean surfaces to hyperbolic knots, Student Mathematical Library 49, Amer. Math. Soc. (2009)
• F Bonahon, H Wong, Quantum traces for representations of surface groups in ${\rm SL}\sb 2(\mathbb C)$, Geom. Topol. 15 (2011) 1569–1615
• B H Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. 77 (1998) 697–736
• R Delgado, Density properties of Euler characteristic-$2$ surface group, $\mathrm{PSL}(2, \mathbb{R})$ character varieties, PhD thesis, University of Maryland (2009) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304923259 {\unhbox0
• B Deroin, N Tholozan, Dominating surface group representations by Fuchsian ones, preprint (2013)
• B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
• R Fricke, F Klein, Vorlesungen über die Theorie der automorphen Funktionen, Vol. I, Johnson Reprint, New York (1965)
• W M Goldman, Discontinuous groups and the Euler class, PhD thesis, Univ. California, Berkeley (1980) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/302998874 {\unhbox0
• W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200–225
• W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557–607
• W M Goldman, Ergodic theory on moduli spaces, Ann. of Math. 146 (1997) 475–507
• W M Goldman, The modular group action on real ${\rm SL}(2)$–characters of a one-holed torus, Geom. Topol. 7 (2003) 443–486
• W M Goldman, Mapping class group dynamics on surface group representations, from: “Problems on mapping class groups and related topics”, (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 189–214
• F Guéritaud, F Kassel, M Wolff, Compact anti-de Sitter $3$–manifolds and folded hyperbolic structures on surfaces, preprint (2013)
• J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176
• R M Kashaev, Coordinates for the moduli space of flat ${\rm PSL}(2,{\mathbb R})$–connections, Math. Res. Lett. 12 (2005) 23–36
• R M Kashaev, On quantum moduli space of flat ${\rm PSL}\sb 2(\mathbb R)$–connections on a punctured surface, from: “Handbook of Teichmüller theory, Vol. I”, (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007) 761–782
• L Keen, C Series, The Riley slice of Schottky space, Proc. London Math. Soc. 69 (1994) 72–90
• S Maloni, F Palesi, S P Tan, On the character varieties of the four-holed sphere to appear in Groups, Geometry, and Dynamics
• J Marché, M Wolff, The modular action on $\mathrm{PSL}\sb 2(\mathbb{R})$–characters in genus $2$, Duke Math. J. 165 (2016) 371–412
• C C Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966) 154–178
• R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299–339
• R C Penner, Weil–Petersson volumes, J. Differential Geom. 35 (1992) 559–608
• J Roger, T Yang, The skein algebra of arcs and links and the decorated Teichmüller space, J. Differential Geom. 96 (2014) 95–140