Actions of Baumslag-Solitar groups on surfaces

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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
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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 and Continuous Dynamical Systems, 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, Prepub. IRMA Lille, 34 (1994), 1-16. doi: 10.1007/PL00004317.
[2]	C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$, Ann. of Math., 129 (1989), 61-69. doi: 10.2307/1971485.
[3]	L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$, Geom. Topol., 8 (2004), 877-924. doi: 10.2140/gt.2004.8.877.
[4]	S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075-1091.
[5]	B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups, Preprint (2001).
[6]	B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups, Duke Math. J., 106 (2001), 581-597. doi: 10.1215/S0012-7094-01-10636-4.
[7]	J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-116. doi: 10.2307/2001018.
[8]	J. Franks and M. Handel, Distortion elements in group actions on surfaces, Duke Math. J., 131 (2006), 441-468. doi: 10.1215/S0012-7094-06-13132-0.
[9]	J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$, Erg. Th. and Dyn. Sys., 27 (2007), 1557-1581. doi: 10.1017/S0143385706001088.
[10]	J. Franks, Handel and K. Parwani, Fixed points of abelian actions, Jour. of Modern Dynamics, 1 (2007), 443-464. doi: 10.3934/jmd.2007.1.443.
[11]	E. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.
[12]	N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$, Algebraic & Geometric Topology, 11 (2011), 1701-1707. doi: 10.2140/agt.2011.11.1701.
[13]	M. Gromov, Asymptotic invariants of infinite groups, in "Geometric Group Theory, London Math. Soc. LNS 182 Cambridge University Press, Cambridge" 2 (1993).
[14]	M. Hirsch, A stable analytic foliation with only exceptional minimal set, Lecture Notes in Math. Springer-Verlag, 468 (1975).
[15]	M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math. Springer-Verlag, Berlin-New York, 583 (1977).
[16]	E. Lima, Common singularities of commuting vector fields on 2-manifolds, Comment. Math. Helv., 39 (1964), 97-110.
[17]	A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups, Proc. Amer. Math. Soc., 138 (2010), 1395-1403. doi: 10.1090/S0002-9939-09-10173-9.
[18]	M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc., 40 (1989), 490-506. doi: 10.1112/jlms/s2-40.3.490.
[19]	Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line, Hokkaido Math. Jour., 23 (1994), 399-422.
[20]	A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13-50. doi: 10.1007/s00574-004-0002-2.
[21]	J. F. Plante, Fixed points of Lie group actions on surfaces, Ergod. Th. and Dynam. Sys., 6 (1986), 149-161. doi: 10.1017/S0143385700003345.
[22]	J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv., 51 (1976), 567-584.
[23]	J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$, Mosc. Math. J., 3 (2003), 123-171.
[24]	M. Shub, Expanding maps, Global Analysis Proceedings of the Symposium on Pure Mathematics, Amer. Math. Soc., Providence, XIV (1970), 273-276.
[25]	M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow, Iteration Theory and Its Functional Equations, Lecture Notes, 1163 (1985), 218-231. doi: 10.1007/BFb0076436.

show all references

References:

[1]	M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes, Preprint, Prepub. IRMA Lille, 34 (1994), 1-16. doi: 10.1007/PL00004317.
[2]	C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$, Ann. of Math., 129 (1989), 61-69. doi: 10.2307/1971485.
[3]	L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$, Geom. Topol., 8 (2004), 877-924. doi: 10.2140/gt.2004.8.877.
[4]	S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075-1091.
[5]	B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups, Preprint (2001).
[6]	B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups, Duke Math. J., 106 (2001), 581-597. doi: 10.1215/S0012-7094-01-10636-4.
[7]	J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-116. doi: 10.2307/2001018.
[8]	J. Franks and M. Handel, Distortion elements in group actions on surfaces, Duke Math. J., 131 (2006), 441-468. doi: 10.1215/S0012-7094-06-13132-0.
[9]	J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$, Erg. Th. and Dyn. Sys., 27 (2007), 1557-1581. doi: 10.1017/S0143385706001088.
[10]	J. Franks, Handel and K. Parwani, Fixed points of abelian actions, Jour. of Modern Dynamics, 1 (2007), 443-464. doi: 10.3934/jmd.2007.1.443.
[11]	E. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.
[12]	N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$, Algebraic & Geometric Topology, 11 (2011), 1701-1707. doi: 10.2140/agt.2011.11.1701.
[13]	M. Gromov, Asymptotic invariants of infinite groups, in "Geometric Group Theory, London Math. Soc. LNS 182 Cambridge University Press, Cambridge" 2 (1993).
[14]	M. Hirsch, A stable analytic foliation with only exceptional minimal set, Lecture Notes in Math. Springer-Verlag, 468 (1975).
[15]	M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math. Springer-Verlag, Berlin-New York, 583 (1977).
[16]	E. Lima, Common singularities of commuting vector fields on 2-manifolds, Comment. Math. Helv., 39 (1964), 97-110.
[17]	A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups, Proc. Amer. Math. Soc., 138 (2010), 1395-1403. doi: 10.1090/S0002-9939-09-10173-9.
[18]	M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc., 40 (1989), 490-506. doi: 10.1112/jlms/s2-40.3.490.
[19]	Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line, Hokkaido Math. Jour., 23 (1994), 399-422.
[20]	A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13-50. doi: 10.1007/s00574-004-0002-2.
[21]	J. F. Plante, Fixed points of Lie group actions on surfaces, Ergod. Th. and Dynam. Sys., 6 (1986), 149-161. doi: 10.1017/S0143385700003345.
[22]	J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv., 51 (1976), 567-584.
[23]	J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$, Mosc. Math. J., 3 (2003), 123-171.
[24]	M. Shub, Expanding maps, Global Analysis Proceedings of the Symposium on Pure Mathematics, Amer. Math. Soc., Providence, XIV (1970), 273-276.
[25]	M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow, Iteration Theory and Its Functional Equations, Lecture Notes, 1163 (1985), 218-231. doi: 10.1007/BFb0076436.

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