American Institute of Mathematical Sciences

Advanced
  2009, 16: 44-55. doi: 10.3934/era.2009.16.44

Research announcement: The structure of groups with a quasiconvex hierarchy

Daniel T. Wise 1,

1. 

Dept. of Math. & Stats., McGill University, Montreal, QC, Canada

Received  August 2009 Revised  September 2009 Published  October 2009

Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. We show that $G$ has a finite index subgroup $G'$ that embeds as a quasiconvex subgroup of a right-angled Artin group. It follows that every quasiconvex subgroup of $G$ is a virtual retract, and is hence separable. The results are applied to certain 3-manifold and one-relator groups.
Keywords: CAT(0) cube complex, right-angled artin group, 3-manifold, subgroup separable, one-relator group..
Mathematics Subject Classification: 53C23, 20F36, 20F55, 20F67, 20F65, 20E2.
Citation: Daniel T. Wise. Research announcement: The structure of groups with a quasiconvex hierarchy. Electronic Research Announcements, 2009, 16: 44-55. doi: 10.3934/era.2009.16.44
[1]

Uri Bader, Jan Dymara. Boundary unitary representations—right-angled hyperbolic buildings. Journal of Modern Dynamics, 2016, 10: 413-437. doi: 10.3934/jmd.2016.10.413
[2]

Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15
[3]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699
[4]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335
[5]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020266
[6]

Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9
[7]

Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39.

[8]

Michael Field, Ian Melbourne, Matthew Nicol, Andrei Török. Statistical properties of compact group extensions of hyperbolic flows and their time one maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 79-96. doi: 10.3934/dcds.2005.12.79
[9]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
[10]

Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
[11]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[12]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281
[13]

Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
[14]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015
[15]

Daniele D'angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda. Schreier graphs of the Basilica group. Journal of Modern Dynamics, 2010, 4 (1) : 167-205. doi: 10.3934/jmd.2010.4.167
[16]

Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130
[17]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139
[18]

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
[19]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[20]

Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (114)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences