## Algebraic & Geometric Topology

### Detecting a subclass of torsion-generated groups

Emily Stark

#### Abstract

We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurable to a group generated by finite-order elements. Consequently, the subclass of groups in this class generated by finite-order elements is not quasi-isometrically rigid. We provide necessary conditions for two groups in this class to be abstractly commensurable. We use these conditions to prove there are infinitely many abstract commensurability classes within each quasi-isometry class of this class that contains a group generated by finite-order elements.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4037-4068.

Dates
Received: 19 November 2017
Revised: 31 July 2018
Accepted: 23 August 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1545102061

Digital Object Identifier
doi:10.2140/agt.2018.18.4037

Mathematical Reviews number (MathSciNet)
MR3892238

Zentralblatt MATH identifier
07006384

#### Citation

Stark, Emily. Detecting a subclass of torsion-generated groups. Algebr. Geom. Topol. 18 (2018), no. 7, 4037--4068. doi:10.2140/agt.2018.18.4037. https://projecteuclid.org/euclid.agt/1545102061

#### References

• D Angluin, Local and global properties in networks of processors, from “Proceedings of the twelfth annual ACM Symposium on Theory of Computing” (R E Miller, S Ginsburg, W A Burkhard, R J Lipton, editors), ACM, New York (1980) 82–93
• J Behrstock, M F Hagen, A Sisto, Quasiflats in hierarchically hyperbolic spaces, preprint (2017)
• J Behrstock, M F Hagen, A Sisto, Thickness, relative hyperbolicity, and randomness in Coxeter groups, Algebr. Geom. Topol. 17 (2017) 705–740
• J A Behrstock, W D Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008) 217–240
• B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145–186
• K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982)
• M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000) 151–194
• P-E Caprace, Buildings with isolated subspaces and relatively hyperbolic Coxeter groups, Innov. Incidence Geom. 10 (2009) 15–31 Correction in 14 (2015) 77–79
• C H Cashen, A Martin, Quasi-isometries between groups with two-ended splittings, Math. Proc. Cambridge Philos. Soc. 162 (2017) 249–291
• A Casson, D Jungreis, Convergence groups and Seifert fibered $3$–manifolds, Invent. Math. 118 (1994) 441–456
• R Charney, K Ruane, N Stambaugh, A Vijayan, The automorphism group of a graph product with no SIL, Illinois J. Math. 54 (2010) 249–262
• R Charney, H Sultan, Contracting boundaries of $\mathrm{CAT}(0)$ spaces, J. Topol. 8 (2015) 93–117
• J Crisp, L Paoluzzi, Commensurability classification of a family of right-angled Coxeter groups, Proc. Amer. Math. Soc. 136 (2008) 2343–2349
• C Cunningham, A Eisenberg, A Piggott, K Ruane, Recognizing right-angled Coxeter groups using involutions, Pacific J. Math. 284 (2016) 41–77
• P Dani, E Stark, A Thomas, Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups, Groups Geom. Dyn. 12 (2018) 1273–1341
• P Dani, A Thomas, Divergence in right-angled Coxeter groups, Trans. Amer. Math. Soc. 367 (2015) 3549–3577
• P Dani, A Thomas, Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, J. Topol. 10 (2017) 1066–1106
• M W Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series 32, Princeton Univ. Press (2008)
• C Dru\commaaccenttu, M Kapovich, Geometric group theory, American Mathematical Society Colloquium Publications 63, Amer. Math. Soc., Providence, RI (2018)
• M J Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449–457
• D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447–510
• P Haïssinsky, L Paoluzzi, G Walsh, Boundaries of Kleinian groups, Illinois J. Math. 60 (2016) 353–364
• M Haulmark, H Nguyen, H C Tran, On boundaries of relatively hyperbolic right-angled Coxeter groups, preprint (2017)
• M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser, Boston, MA (2001)
• J-F Lafont, Diagram rigidity for geometric amalgamations of free groups, J. Pure Appl. Algebra 209 (2007) 771–780
• F T Leighton, Finite common coverings of graphs, J. Combin. Theory Ser. B 33 (1982) 231–238
• I Levcovitz, A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups, preprint (2017)
• I Levcovitz, Divergence of $\mathrm{CAT}(0)$ cube complexes and Coxeter groups, Algebr. Geom. Topol. 18 (2018) 1633–1673
• W Malone, Topics in geometric group theory, PhD thesis, The University of Utah (2010) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/305212500 {\unhbox0
• P Papasoglu, K Whyte, Quasi-isometries between groups with infinitely many ends, Comment. Math. Helv. 77 (2002) 133–144
• J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994)
• E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997) 53–109
• P Scott, T Wall, Topological methods in group theory, from “Homological group theory” (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137–203
• E Stark, Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams, Geom. Dedicata 186 (2017) 39–74
• P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1–54