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May  2015, 35(5): 1817-1827. doi: 10.3934/dcds.2015.35.1817

Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements

Juan Alonso 1, , Nancy Guelman 2, and  Juliana Xavier 2,

1. 

Centro de Matemática, Facultad de Ciencias, Iguá 4225, Montevideo, CP 11400, Uruguay
2. 

IMERL, Facultad de Ingeniería, Julio Herrera y Reissig 565, Montevideo, CP 11300, Uruguay, Uruguay

Received  March 2014 Revised  September 2014 Published  December 2014

Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces of genus $g\geq 1$ with (pseudo)-Anosov elements. That is, we consider a closed surface $S$ of genus $g\geq 1$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is (pseudo)-Anosov with stretch factor $\lambda >1$. We show that $\langle f,h \rangle$ is not a faithful representation of $BS(1, n)$ if $\lambda > n$. We also show that there are no faithful representations of $BS(1, n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $\lambda$).
Keywords: Rigidity theory., Baumslag-Solitar groups, group actions, pseudo-Anosov homeomorphisms, Surface homeomorphisms.
Mathematics Subject Classification: Primary: 37C85, 37E30, 37B05, 37E4.
Citation: 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
References:
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G. Baumslag and D. Solitar, Some two generator one-relator non-Hopfian groups,, Bull. Amer. Math. Soc., 68 (1962), 199.  doi: 10.1090/S0002-9904-1962-10745-9.  Google Scholar

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B. Farb, A. Lubotzky and Y. Minsky, Rank one phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581.  doi: 10.1215/S0012-7094-01-10636-4.  Google Scholar

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B. Farb and L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups,, Invent. Math., 131 (1998), 419.  doi: 10.1007/s002220050210.  Google Scholar

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N. Guelman and I. Liousse, C1- actions of Baumslag-Solitar groups on S1,, AGT, 11 (2011), 1701.  doi: 10.2140/agt.2011.11.1701.  Google Scholar

[9]

N. Guelman and I. Liousse, Actions of Baumslag-Solitar groups on surfaces,, Disc. Cont. Dyn. Sys., 33 (2013), 1945.   Google Scholar

[10]

M. E. Hamstrom, Homotopy groups of the space of homeomorphisms on a $2$- manifold,, Ill. J. Math., 10 (1996), 563.   Google Scholar

[11]

A. Hatcher, Algebraic Topology,, Cambridge University Press, (2002).   Google Scholar

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A. Koropecki and F. Tal, Bounded and unbounded behaviour for rational pseudo rotations,, Preprint, ().   Google Scholar

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J. D. McCarthy, Normalizers and centralizers of pseudo-Anosov mapping classes,, Preprint., ().   Google Scholar

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A. Navas, Groupes resolubles de diffeomorphismes 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

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J. F. Plante, Solvable groups acting on the line,, Trans. Amer. Math. Soc., 278 (1983), 401.  doi: 10.1090/S0002-9947-1983-0697084-7.  Google Scholar

[16]

J. Palis and J. C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori,, Ann. Sc. ENS, 22 (1989), 99.   Google Scholar

[17]

J. Rocha, A note on the $C 0$-centralizer of an open class of bidimensional Anosov diffeomorphisms,, Aequ. math., 76 (2008), 105.  doi: 10.1007/s00010-007-2910-x.  Google Scholar

[18]

R. Zimmer, Actions of semisimple groups and discrete subgroups,, Proc. Internat. Congr. Math., 2 (1987), 1247.   Google Scholar

show all references

References:
[1]

M. Bestvina, Questions in geometric group theory,, Available from , ().   Google Scholar

[2]

G. Baumslag and D. Solitar, Some two generator one-relator non-Hopfian groups,, Bull. Amer. Math. Soc., 68 (1962), 199.  doi: 10.1090/S0002-9904-1962-10745-9.  Google Scholar

[3]

D. Fisher, Groups acting on manifolds: Around the Zimmer program,, Geometry, (2011), 72.  doi: 10.7208/chicago/9780226237909.001.0001.  Google Scholar

[4]

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

[5]

B. Farb, A. Lubotzky and Y. Minsky, Rank one phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581.  doi: 10.1215/S0012-7094-01-10636-4.  Google Scholar

[6]

B. Farb and D. Margalit, A Primer on Mapping Class Groups,, {Princeton University Press}, (2012).   Google Scholar

[7]

B. Farb and L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups,, Invent. Math., 131 (1998), 419.  doi: 10.1007/s002220050210.  Google Scholar

[8]

N. Guelman and I. Liousse, C1- actions of Baumslag-Solitar groups on S1,, AGT, 11 (2011), 1701.  doi: 10.2140/agt.2011.11.1701.  Google Scholar

[9]

N. Guelman and I. Liousse, Actions of Baumslag-Solitar groups on surfaces,, Disc. Cont. Dyn. Sys., 33 (2013), 1945.   Google Scholar

[10]

M. E. Hamstrom, Homotopy groups of the space of homeomorphisms on a $2$- manifold,, Ill. J. Math., 10 (1996), 563.   Google Scholar

[11]

A. Hatcher, Algebraic Topology,, Cambridge University Press, (2002).   Google Scholar

[12]

A. Koropecki and F. Tal, Bounded and unbounded behaviour for rational pseudo rotations,, Preprint, ().   Google Scholar

[13]

J. D. McCarthy, Normalizers and centralizers of pseudo-Anosov mapping classes,, Preprint., ().   Google Scholar

[14]

A. Navas, Groupes resolubles de diffeomorphismes 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

[15]

J. F. Plante, Solvable groups acting on the line,, Trans. Amer. Math. Soc., 278 (1983), 401.  doi: 10.1090/S0002-9947-1983-0697084-7.  Google Scholar

[16]

J. Palis and J. C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori,, Ann. Sc. ENS, 22 (1989), 99.   Google Scholar

[17]

J. Rocha, A note on the $C 0$-centralizer of an open class of bidimensional Anosov diffeomorphisms,, Aequ. math., 76 (2008), 105.  doi: 10.1007/s00010-007-2910-x.  Google Scholar

[18]

R. Zimmer, Actions of semisimple groups and discrete subgroups,, Proc. Internat. Congr. Math., 2 (1987), 1247.   Google Scholar

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