## Algebraic & Geometric Topology

### On the coarse geometry of certain right-angled Coxeter groups

#### Abstract

Let $Γ$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $Γ$ is $CℱS$, we prove that the right-angled Coxeter group $GΓ$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these groups. Furthermore, we prove that $GΓ$ is hyperbolic relative to a collection of $CℱS$ right-angled Coxeter subgroups of $GΓ$. Consequently, the divergence of $GΓ$ is linear, quadratic or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high-dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex, torsion-free, infinite-index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 6 (2019), 3075-3118.

Dates
Revised: 31 December 2018
Accepted: 3 March 2019
First available in Project Euclid: 29 October 2019

https://projecteuclid.org/euclid.agt/1572314550

Digital Object Identifier
doi:10.2140/agt.2019.19.3075

Mathematical Reviews number (MathSciNet)
MR4023336

Zentralblatt MATH identifier
07142626

#### Citation

Nguyen, Hoang Thanh; Tran, Hung Cong. On the coarse geometry of certain right-angled Coxeter groups. Algebr. Geom. Topol. 19 (2019), no. 6, 3075--3118. doi:10.2140/agt.2019.19.3075. https://projecteuclid.org/euclid.agt/1572314550

#### References

• J Behrstock, A counterexample to questions about boundaries, stability, and commensurability, preprint (2017)
• J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339–356
• J Behrstock, C Dru\commaaccenttu, Divergence, thick groups, and short conjugators, Illinois J. Math. 58 (2014) 939–980
• J Behrstock, C Dru\commaaccenttu, L Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann. 344 (2009) 543–595
• 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, T Januszkiewicz, W D Neumann, Quasi-isometric classification of some high dimensional right-angled Artin groups, Groups Geom. Dyn. 4 (2010) 681–692
• J A Behrstock, W D Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008) 217–240
• P-E Caprace, Buildings with isolated subspaces and relatively hyperbolic Coxeter groups, Innov. Incidence Geom. 10 (2009) 15–31 Corrrection in 14 (2015) 77–79
• R Charney, H Sultan, Contracting boundaries of $\rm CAT(0)$ spaces, J. Topol. 8 (2015) 93–117
• M Cordes, Morse boundaries of proper geodesic metric spaces, Groups Geom. Dyn. 11 (2017) 1281–1306
• M Cordes, D Hume, Stability and the Morse boundary, J. Lond. Math. Soc. 95 (2017) 963–988
• 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)
• M W Davis, T Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000) 229–235
• M W Davis, B Okun, Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups, Geom. Topol. 5 (2001) 7–74
• C Dru\commaaccenttu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959–1058
• M G Durham, S J Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15 (2015) 2839–2859
• A Genevois, Hyperbolicities in $\mathrm{CAT}(0)$ cube complexes, preprint (2017)
• S M Gersten, Divergence in $3$–manifold groups, Geom. Funct. Anal. 4 (1994) 633–647
• S M Gersten, Quadratic divergence of geodesics in ${\rm CAT}(0)$ spaces, Geom. Funct. Anal. 4 (1994) 37–51
• C M Gordon, Artin groups, $3$–manifolds and coherence, Bol. Soc. Mat. Mexicana 10 (2004) 193–198
• M Haulmark, H T Nguyen, H C Tran, On the relative hyperbolicity and manifold structure of certain right-angled Coxeter groups, preprint (2017)
• M Kapovich, B Leeb, $3$–Manifold groups and nonpositive curvature, Geom. Funct. Anal. 8 (1998) 841–852
• H Kim, Stable subgroups and Morse subgroups in mapping class groups, Internat. J. Algebra Comput. 29 (2019) 893–903
• I Levcovitz, Divergence of $\rm CAT(0)$ cube complexes and Coxeter groups, Algebr. Geom. Topol. 18 (2018) 1633–1673
• J Russell, D Spriano, H C Tran, Convexity in hierarchically hyperbolic spaces, preprint (2018)
• A Sisto, On metric relative hyperbolicity, preprint (2012)
• H C Tran, Malnormality and join-free subgroups in right-angled Coxeter groups, preprint (2017)
• H C Tran, On strongly quasiconvex subgroups, Geom. Topol. 23 (2019) 1173–1235