## Algebraic & Geometric Topology

### $C^1$–actions of Baumslag–Solitar groups on $S^1$

#### Abstract

Let $BS(1,n)=〈a,b∣aba−1=bn〉$ be the solvable Baumslag–Solitar group, where $n≥2$. It is known that $BS(1,n)$ is isomorphic to the group generated by the two affine maps of the line: $f0(x)=x+1$ and $h0(x)=nx$. The action on $S1=ℝ∪∞$ generated by these two affine maps $f0$ and $h0$ is called the standard affine one. We prove that any faithful representation of $BS(1,n)$ into $Diff1(S1)$ is semiconjugated (up to a finite index subgroup) to the standard affine action.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1701-1707.

Dates
Received: 28 October 2010
Revised: 6 April 2011
Accepted: 9 April 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715241

Digital Object Identifier
doi:10.2140/agt.2011.11.1701

Mathematical Reviews number (MathSciNet)
MR2821437

Zentralblatt MATH identifier
1221.37048

#### Citation

Guelman, Nancy; Liousse, Isabelle. $C^1$–actions of Baumslag–Solitar groups on $S^1$. Algebr. Geom. Topol. 11 (2011), no. 3, 1701--1707. doi:10.2140/agt.2011.11.1701. https://projecteuclid.org/euclid.agt/1513715241

#### References

• L Burslem, A Wilkinson, Global rigidity of solvable group actions on $S\sp 1$, Geom. Topol. 8 (2004) 877–924
• J Cantwell, L Conlon, An interesting class of $C\sp 1$ foliations, Topology Appl. 126 (2002) 281–297
• B Farb, J Franks, Groups of homeomorphisms of one manifolds I: Actions of nonlinear groups
• É Ghys, Groups acting on the circle, Enseign. Math. $(2)$ 47 (2001) 329–407
• Y Moriyama, Polycyclic groups of diffeomorphisms on the half-line, Hokkaido Math. J. 23 (1994) 399–422
• A Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite, Bull. Braz. Math. Soc. $($N.S.$)$ 35 (2004) 13–50
• A Navas, A finitely generated, locally indicable group with no faithful action by $C\sp 1$ diffeomorphisms of the interval, Geom. Topol. 14 (2010) 573–584
• A Navas, On the dynamics of (left) orderable groups, Ann. Inst. Fourier $($Grenoble$)$ 60 (2010) 1685–1740
• J F Plante, Solvable groups acting on the line, Trans. Amer. Math. Soc. 278 (1983) 401–414
• J C Rebelo, R R Silva, The multiple ergodicity of nondiscrete subgroups of ${\rm Diff}\sp \omega(S\sp 1)$, Mosc. Math. J. 3 (2003) 123–171, 259
• C Rivas, On spaces of Conradian group orderings, J. Group Theory 13 (2010) 337–353