Abstract
In this chapter, we study deformations of (possibly indiscrete) faithful representations of Fuchsian groups such that almost all points on the deformations are still faithful. Let L be a splittable Fuchsian group, which includes all Fuchsian groups with Euler characteristic at most − 1; see Definition 4.3. We will prove that an arbitrarily small deformation of a given representation can be chosen so that the new trace spectrum is almost disjoint from the original one (Theorem 4.1). Then we show X proj(L) contains at least one indiscrete representation (Lemma 4.10). Moreover, if an open set U contains at least one indiscrete representation in X proj(L), then U contains uncountably many pairwise inequivalent indiscrete representation in X proj(L) (Theorem 4.2).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
J. Barlev, T. Gelander, Compactifications and algebraic completions of limit groups. J. Anal. Math. 112, 261–287 (2010). MR 2763002
M. Bestvina, M. Feighn, Notes on Sela’s work: limit groups and Makanin-Razborov diagrams, in Geometric and Cohomological Methods in Group Theory. London Mathematical Society Lecture Note Series, vol. 358 (Cambridge University Press, Cambridge, 2009), pp. 1–29. MR 2605174
A. Borel, On free subgroups of semisimple groups. Enseign. Math. (2) 29(1–2), 151–164 (1983). MR 702738
K. Bou-Rabee, M. Larsen, Linear groups with Borel’s property. J. Eur. Math. Soc. 19(5), 1293–1330 (2017). MR 3635354
E. Breuillard, D. Guralnick, B. Green, T. Tao, Strongly dense free subgroups of semisimple algebraic groups. Israel J. Math. 192(1), 347–379 (2012)
W.M. Goldman, Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988). MR 952283
K. Mann, Rigidity and flexibility of group actions on the circle, in Handbook of Group Actions (2015, to appear)
K. Mann, Spaces of surface group representations. Invent. Math. 201(2), 669–710 (2015). MR 3370623
S. Matsumoto, Some remarks on foliated S 1 bundles. Invent. Math. 90(2), 343–358 (1987). MR 910205
H. Wilton, Solutions to Bestvina & Feighn’s exercises on limit groups, in Geometric and Cohomological Methods in Group Theory. London Mathematical Society Lecture Note Series, vol. 358 (Cambridge University Press, Cambridge, 2009), pp. 30–62. MR 2605175 (2011g:20037)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kim, Sh., Koberda, T., Mj, M. (2019). Splittable Fuchsian Groups. In: Flexibility of Group Actions on the Circle. Lecture Notes in Mathematics, vol 2231. Springer, Cham. https://doi.org/10.1007/978-3-030-02855-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-02855-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02854-1
Online ISBN: 978-3-030-02855-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)