## Duke Mathematical Journal

### Quasi-isometric classification of graph manifold groups

#### Abstract

We show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi-isometric. We also classify the quasi-isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite two-colored graphs. A corollary is the quasi-isometric classification of Artin groups whose presentation graphs are trees. In particular, any two right-angled Artin groups whose presentation graphs are trees of diameter greater than $2$ are quasi-isometric; further, this quasi-isometry class does not include any other right-angled Artin groups

#### Article information

Source
Duke Math. J., Volume 141, Number 2 (2008), 217-240.

Dates
First available in Project Euclid: 17 January 2008

https://projecteuclid.org/euclid.dmj/1200601791

Digital Object Identifier
doi:10.1215/S0012-7094-08-14121-3

Mathematical Reviews number (MathSciNet)
MR2376814

Zentralblatt MATH identifier
1194.20045

#### Citation

Behrstock, Jason A.; Neumann, Walter D. Quasi-isometric classification of graph manifold groups. Duke Math. J. 141 (2008), no. 2, 217--240. doi:10.1215/S0012-7094-08-14121-3. https://projecteuclid.org/euclid.dmj/1200601791

#### References

• J. Behrstock, C. DruţU, and L. Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, preprint,\arxivmath/0512592v4 [math.GT]
• M. Bestvina, personal communication, Oct. 2005.
• N. Brady, J. P. Mccammond, B. MüHlherr, and W. D. Neumann, Rigidity of Coxeter groups and Artin groups'' in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, Israel, 2000), Geom. Dedicata 94 (2002), 91--109.
• A. M. Brunner, Geometric quotients of link groups, Topology Appl. 48 (1992), 245--262.
• J. W. Cannon and D. Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992), 419--431.
• D. Eisenbud and W. D. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Stud. 110, Princeton Univ. Press, Princeton, 1985.
• A. Eskin, D. Fisher, and K. Whyte, Quasi-isometries and rigidity of solvable groups, preprint,\arxivmath/0511647v3 [math.GR]
• S. M. Gersten, Divergence in $3$-manifold groups, Geom. Funct. Anal. 4 (1994), 633--647.
• C. Mca. Gordon, Artin groups, $3$-manifolds and coherence, Bol. Soc. Mat. Mexicana (3) 10 (2004), 193--198.
• M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53--73.
• —, Asymptotic invariants of infinite groups'' in Geometric Group Theory, Vol. 2 (Sussex, U.K. 1991) London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, 1993, 1--295.
• S. M. Hermiller and J. Meier, Artin groups, rewriting systems and three-manifolds, J. Pure Appl. Algebra 136 (1999), 141--156.
• M. Kapovich and B. Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math. 128 (1997), 393--416.
• —, $3$-manifold groups and nonpositive curvature, Geom. Funct. Anal. 8 (1998), 841--852.
• B. D. Mckay, nauty, http://cs.anu.edu.au/$\sim$bdm/nauty/
• J. Milnor, A note on curvature and the fundamental group, J. Differential Geometry 2 (1968), 1--7.
• J. W. Morgan and H. Bass, eds., The Smith Conjecture (New York, 1979), Pure Appl. Math. 112, Academic Press, Orlando, 1984.
• W. D. Neumann, Commensurability and virtual fibration for graph manifolds, Topology 36 (1997), 355--378.
• W. D. Neumann and G. A. Swarup, Canonical decompositions of $3$-manifolds, Geom. Topol. 1 (1997), 21--40.
• P. Papasoglu and K. Whyte, Quasi-isometries between groups with infinitely many ends, Comment. Math. Helv. 77 (2002), 133--144.
• G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint,\arxivmath/0211159v1 [math.DG]
• —, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint,\arxivmath/0307245v1 [math.DG]
• —, Ricci flow with surgery on three-manifolds, preprint,\arxivmath/0303109v1 [math.DG]
• E. G. Rieffel, Groups quasi-isometric to $\mathbf H^2\times\mathbf R$, J. London Math. Soc. (2) 64 (2001), 44--60.
• R. E. Schwartz, The quasi-isometry classification of rank one lattices, Inst. Hautes Études Sci. Publ. Math. 82 (1995), 133--168.
• G. P. Scott, Finitely generated $3$-manifold groups are finitely presented, J. London Math. Soc. (2) 6 (1973), 437--440.
• K. Shan, personal communication, March 2006.
• A. S. šVarc, A volume invariant of coverings (in Russian), Dokl. Akad. Nauk. SSSR (N.S.) 105 (1955), 32--34.
• W. P. Thurston, Hyperbolic structures on $3$-manifolds, I: Deformation of a cylindrical manifold, Ann. of Math. (2) 124 (1986), 203--246.