Actions of Baumslag-Solitar groups on surfaces

Previous Article
Stochastic perturbations and Ulam's method for W-shaped maps
DCDS Home
This Issue
Next Article
Phase transitions in one-dimensional subshifts

May 2013, 33(5): 1945-1964. doi: 10.3934/dcds.2013.33.1945

Actions of Baumslag-Solitar groups on surfaces

Nancy Guelman ^1, and Isabelle Liousse ^2,

1.	IMERL, Facultad de Ingeniería, Universidad de La República, C.C. 30,Montevideo
2.	Laboratoire Paul PAINLEVÉ, Université de Lille1, 59655 Villeneuve d'Ascq Cédex, France

Received November 2011 Revised July 2012 Published December 2012

Abstract
Related Papers

Let $BS(1, n) =< a, b \ | \ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that $BS(1, n)$ is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $.
This paper deals with the dynamics of actions of $BS(1, n)$ on closed orientable surfaces. We exhibit a smooth $BS(1,n)$-action without finite orbits on $\mathbb{T} ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid.
We develop a general dynamical study for faithful topological $BS(1,n)$-actions on closed surfaces $S$. We prove that such actions $ < f, h \ | \ h o f o h^{-1} = f^n >$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty.
When $S= \mathbb{T}^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of $BS(1,n)$ on $\mathbb{T}^2$.
When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ and isotopic to identity then $fix(f)$ contains any minimal set.

Keywords: Baumslag Solitar group, Actions on surfaces, minimal sets..

Mathematics Subject Classification: Primary: 37C85, 37E30; Secondary: 37B0.

Citation: Nancy Guelman, Isabelle Liousse. Actions of Baumslag-Solitar groups on surfaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1945-1964. doi: 10.3934/dcds.2013.33.1945

References:

[1]	M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes,, Preprint, 34 (1994), 1. doi: 10.1007/PL00004317. Google Scholar
[2]	C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$,, Ann. of Math., 129 (1989), 61. doi: 10.2307/1971485. Google Scholar
[3]	L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$,, Geom. Topol., 8 (2004), 877. doi: 10.2140/gt.2004.8.877. Google Scholar
[4]	S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075. Google Scholar
[5]	B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups,, Preprint (2001)., (2001). Google Scholar
[6]	B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581. doi: 10.1215/S0012-7094-01-10636-4. Google Scholar
[7]	J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107. doi: 10.2307/2001018. Google Scholar
[8]	J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441. doi: 10.1215/S0012-7094-06-13132-0. Google Scholar
[9]	J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$,, Erg. Th. and Dyn. Sys., 27 (2007), 1557. doi: 10.1017/S0143385706001088. Google Scholar
[10]	J. Franks, Handel and K. Parwani, Fixed points of abelian actions,, Jour. of Modern Dynamics, 1 (2007), 443. doi: 10.3934/jmd.2007.1.443. Google Scholar
[11]	E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329. Google Scholar
[12]	N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$,, Algebraic & Geometric Topology, 11 (2011), 1701. doi: 10.2140/agt.2011.11.1701. Google Scholar
[13]	M. Gromov, Asymptotic invariants of infinite groups,, in, 2 (1993). Google Scholar
[14]	M. Hirsch, A stable analytic foliation with only exceptional minimal set,, Lecture Notes in Math. Springer-Verlag, 468 (1975). Google Scholar
[15]	M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Math. Springer-Verlag, 583 (1977). Google Scholar
[16]	E. Lima, Common singularities of commuting vector fields on 2-manifolds,, Comment. Math. Helv., 39 (1964), 97. Google Scholar
[17]	A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups,, Proc. Amer. Math. Soc., 138 (2010), 1395. doi: 10.1090/S0002-9939-09-10173-9. Google Scholar
[18]	M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori,, J. London Math. Soc., 40 (1989), 490. doi: 10.1112/jlms/s2-40.3.490. Google Scholar
[19]	Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line,, Hokkaido Math. Jour., 23 (1994), 399. Google Scholar
[20]	A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite,, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13. doi: 10.1007/s00574-004-0002-2. Google Scholar
[21]	J. F. Plante, Fixed points of Lie group actions on surfaces,, Ergod. Th. and Dynam. Sys., 6 (1986), 149. doi: 10.1017/S0143385700003345. Google Scholar
[22]	J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations,, Comment. Math. Helv., 51 (1976), 567. Google Scholar
[23]	J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$,, Mosc. Math. J., 3 (2003), 123. Google Scholar
[24]	M. Shub, Expanding maps,, Global Analysis Proceedings of the Symposium on Pure Mathematics, XIV (1970), 273. Google Scholar
[25]	M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow,, Iteration Theory and Its Functional Equations, 1163 (1985), 218. doi: 10.1007/BFb0076436. Google Scholar

show all references

References:

[1]	M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes,, Preprint, 34 (1994), 1. doi: 10.1007/PL00004317. Google Scholar
[2]	C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$,, Ann. of Math., 129 (1989), 61. doi: 10.2307/1971485. Google Scholar
[3]	L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$,, Geom. Topol., 8 (2004), 877. doi: 10.2140/gt.2004.8.877. Google Scholar
[4]	S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075. Google Scholar
[5]	B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups,, Preprint (2001)., (2001). Google Scholar
[6]	B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581. doi: 10.1215/S0012-7094-01-10636-4. Google Scholar
[7]	J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107. doi: 10.2307/2001018. Google Scholar
[8]	J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441. doi: 10.1215/S0012-7094-06-13132-0. Google Scholar
[9]	J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$,, Erg. Th. and Dyn. Sys., 27 (2007), 1557. doi: 10.1017/S0143385706001088. Google Scholar
[10]	J. Franks, Handel and K. Parwani, Fixed points of abelian actions,, Jour. of Modern Dynamics, 1 (2007), 443. doi: 10.3934/jmd.2007.1.443. Google Scholar
[11]	E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329. Google Scholar
[12]	N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$,, Algebraic & Geometric Topology, 11 (2011), 1701. doi: 10.2140/agt.2011.11.1701. Google Scholar
[13]	M. Gromov, Asymptotic invariants of infinite groups,, in, 2 (1993). Google Scholar
[14]	M. Hirsch, A stable analytic foliation with only exceptional minimal set,, Lecture Notes in Math. Springer-Verlag, 468 (1975). Google Scholar
[15]	M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Math. Springer-Verlag, 583 (1977). Google Scholar
[16]	E. Lima, Common singularities of commuting vector fields on 2-manifolds,, Comment. Math. Helv., 39 (1964), 97. Google Scholar
[17]	A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups,, Proc. Amer. Math. Soc., 138 (2010), 1395. doi: 10.1090/S0002-9939-09-10173-9. Google Scholar
[18]	M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori,, J. London Math. Soc., 40 (1989), 490. doi: 10.1112/jlms/s2-40.3.490. Google Scholar
[19]	Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line,, Hokkaido Math. Jour., 23 (1994), 399. Google Scholar
[20]	A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite,, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13. doi: 10.1007/s00574-004-0002-2. Google Scholar
[21]	J. F. Plante, Fixed points of Lie group actions on surfaces,, Ergod. Th. and Dynam. Sys., 6 (1986), 149. doi: 10.1017/S0143385700003345. Google Scholar
[22]	J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations,, Comment. Math. Helv., 51 (1976), 567. Google Scholar
[23]	J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$,, Mosc. Math. J., 3 (2003), 123. Google Scholar
[24]	M. Shub, Expanding maps,, Global Analysis Proceedings of the Symposium on Pure Mathematics, XIV (1970), 273. Google Scholar
[25]	M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow,, Iteration Theory and Its Functional Equations, 1163 (1985), 218. doi: 10.1007/BFb0076436. Google Scholar

[1]	Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817
[2]	Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.
[3]	Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[4]	José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178
[5]	Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301
[6]	A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.
[7]	Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[8]	Francisco Brito, Maria Luiza Leite, Vicente de Souza Neto. Liouville's formula under the viewpoint of minimal surfaces. Communications on Pure & Applied Analysis, 2004, 3 (1) : 41-51. doi: 10.3934/cpaa.2004.3.41
[9]	Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[10]	Francesco Maggi, Salvatore Stuvard, Antonello Scardicchio. Soap films with gravity and almost-minimal surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-36. doi: 10.3934/dcds.2019236
[11]	Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483
[12]	Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765
[13]	Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041
[14]	Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114
[15]	Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
[16]	Ronald A. Knight. Compact minimal sets in continuous recurrent flows. Conference Publications, 1998, 1998 (Special) : 397-407. doi: 10.3934/proc.1998.1998.397
[17]	Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365
[18]	Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[19]	Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
[20]	Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences