American Institute of Mathematical Sciences

Advanced
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:
[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

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

[1]

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
[2]

Światosław R. Gal, Jarek Kędra. On distortion in groups of homeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 609-622. doi: 10.3934/jmd.2011.5.609
[3]

Chris Johnson, Martin Schmoll. Pseudo-Anosov eigenfoliations on Panov planes. Electronic Research Announcements, 2014, 21: 89-108. doi: 10.3934/era.2014.21.89
[4]

Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511
[5]

Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239
[6]

S. Öykü Yurttaş. Dynnikov and train track transition matrices of pseudo-Anosov braids. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 541-570. doi: 10.3934/dcds.2016.36.541
[7]

Francisco R. Ruiz del Portal. Stable sets of planar homeomorphisms with translation pseudo-arcs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2379-2390. doi: 10.3934/dcdss.2019149
[8]

Jingzhi Yan. Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4571-4602. doi: 10.3934/dcds.2018200
[9]

Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957
[10]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[11]

Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193
[12]

Hieu Trung Do, Thomas A. Schmidt. New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant. Journal of Modern Dynamics, 2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541
[13]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213
[14]

Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154.

[15]

Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291
[16]

Salvador Addas-Zanata, Fábio A. Tal. Homeomorphisms of the annulus with a transitive lift II. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 651-668. doi: 10.3934/dcds.2011.31.651
[17]

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
[18]

Zhenqi Jenny Wang. New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions. Journal of Modern Dynamics, 2010, 4 (4) : 585-608. doi: 10.3934/jmd.2010.4.585
[19]

Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191
[20]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences