# Spaces of surface group representations

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## Abstract

Let \(\Gamma _g\) denote the fundamental group of a closed surface of genus \(g \ge 2\). We show that every geometric representation of \(\Gamma _g\) into the group of orientation-preserving homeomorphisms of the circle is *rigid*, meaning that its deformations form a single semi-conjugacy class. As a consequence, we give a new lower bound on the number of topological components of the space of representations of \(\Gamma _g\) into \({{\mathrm{Homeo}}}_+(S^1)\). Precisely, for each nontrivial divisor \(k\) of \(2g-2\), there are at least \(|k|^{2g} + 1\) components containing representations with Euler number \(\frac{2g-2}{k}\). Our methods apply to representations of surface groups into finite covers of \({{\mathrm{PSL}}}(2,\mathbb {R})\) and into \({{\mathrm{Diff}}}_+(S^1)\) as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of Calegari–Walker. This is a new approach to studying deformation classes of group actions on the circle, and may be of independent interest.

## Notes

### Acknowledgments

The author thanks Danny Calegari for suggesting that rotation numbers might distinguish components of \({{\mathrm{Hom}}}(\Gamma _g, {{\mathrm{Homeo}}}_+(S^1))\), and explaining the philosophy of rotation numbers as “trace coordinates” on representation spaces. She also thanks Jonathan Bowden, Benson Farb, Shigenori Matsumoto, and Alden Walker for many helpful conversations and suggestions regarding this work, and to the referee for a careful reading and thoughtful input.

## References

- 1.Bowden, J.: Contact structures, deformations and taut foliations (preprint). arxiv:1304.3833v1
- 2.Calegari, D., Dunfield, N.: Laminations and groups of homeomorphisms of the circle. Invent. Math.
**152**(1), 149–204 (2003)MathSciNetCrossRefGoogle Scholar - 3.Calegari, D., Walker, A.: Ziggurats and rotation numbers. J. Mod. Dyn.
**5**(4), 711–746 (2011)MathSciNetGoogle Scholar - 4.Ghys, E.: Groups acting on the circle. L’Enseignement Mathématique
**47**, 329–407 (2001)MathSciNetGoogle Scholar - 5.Ghys, E.: Groupes d’homéomorphismes du cercle et cohomologie bornée. In: The Lefschetz Centennial Conference, Part III. Contemporary Mathematics, vol. 58 III, pp. 81–106. American Mathematical Society, Providence (1987)Google Scholar
- 6.Goldman, W.: Topological components of spaces of representations. Invent. Math.
**93**(3), 557–607 (1998)CrossRefGoogle Scholar - 7.Milnor, J.: On the existence of a connection with curvature zero. Commet. Math. Helv.
**32**(1), 215–223 (1958)MathSciNetCrossRefGoogle Scholar - 8.Matsumoto, S.: Numerical invariants for semi-conjugacy of homeomorphisms of the circle. Proc. AMS
**96**(1), 163–168 (1986)CrossRefGoogle Scholar - 9.Matsumoto, S.: Some remarks on foliated \(S^1\) bundles. Invent. math.
**90**, 343–358 (1987)MathSciNetCrossRefGoogle Scholar - 10.Navas, A.: Groups of Circle Diffeomorphisms. University of Chicago press, Chicago (2011)CrossRefGoogle Scholar
- 11.Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. (2)
**17**(3), 555–565 (1978)CrossRefGoogle Scholar - 12.Wood, J.: Bundles with totally disconnected structure group. Comment. Math. Helv.
**51**, 183–199 (1971)Google Scholar