# Dehn functions of subgroups of right-angled Artin groups

- 21 Downloads

## Abstract

We show that for each positive integer *k* there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as \(n^k\). As a consequence we produce examples of right-angled Artin groups containing finitely presented subgroups whose Dehn functions grow as \(n^{k+2}\).

## Keywords

Dehn functions Right-angled Artin groups Free-by-cyclic groups Special cube complexes Growth of automorphisms## Mathematics Subject Classification (2010)

Primary 20F65 20E05 20F67 57M20## Notes

### Acknowledgements

Noel Brady acknowledges support from NSF grant DMS 0906962 and Simons Foundation award 430097. Ignat Soroko also acknowledges support from Simons Foundation award 430097 and from the research assistantship of Dr. Jing Tao (NSF grant DMS 1611758 and NSF Career grant DMS 1651963). Both authors are grateful to Mark Hagen and Daniel Woodhouse for discussions about Gersten’s group. Finally, both authors are very grateful to the referee for valuable comments and suggestions which improved the quality of this paper.

## References

- 1.Agol, I.: Criteria for virtual fibering. J. Topol.
**1**(2), 269–284 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Agol, I.: The virtual Haken conjecture. With an appendix by Ian Agol, Daniel Groves, and Jason Manning. Doc. Math.
**18**, 1045–1087 (2013)MathSciNetzbMATHGoogle Scholar - 3.Barnard, J., Brady, N.: Distortion of surface groups in \(\text{ CAT }(0)\) free-by-cyclic groups. Geom. Dedicata
**120**, 119–139 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math.
**129**(3), 445–470 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bieri, R.: Normal subgroups in duality groups and in groups of cohomological dimension 2. J. Pure Appl. Algebra
**7**(1), 35–51 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Birget, J., Ol’shanskii, A., Rips, E., Sapir, M.: Isomperimetric functions of groups and computational complexity of the word problem. Ann. Math. (2)
**156**(2), 476–518 (2002)CrossRefGoogle Scholar - 7.Brady, N., Forester, M.: Snowflake geometry in \(\text{ CAT }(0)\) groups. J. Topol.
**10**, 883–920 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Brady, N.: Dehn Functions and Non-Positive Curvature. The Geometry of the Word Problem for Finitely Generated Groups, pp. 1–79. Birkhäuser, Basel (2007)Google Scholar
- 9.Brady, N., Bridson, M.: There is only one gap in the isoperimetric spectrum. Geom. Funct. Anal.
**10**(5), 1053–1070 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Bridson, M.R.: Combings of semidirect products and \(3\)-manifold groups. Geom. Funct. Anal.
**3**(3), 263–278 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Bridson, M.R.: The geometry of the word problem. Invitations to geometry and topology, 29–91, Oxf. Grad. Texts Math., 7, Oxford University Press, Oxford (2002)Google Scholar
- 12.Bridson, M.R.: On the subgroups of right-angled Artin groups and mapping class groups. Math. Res. Lett.
**20**(2), 203–212 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Bridson, M.R.: Polynomial Dehn functions and the length of asynchronously automatic structures. Proc. Lond. Math Soc. (3)
**85**, 441–466 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Bridson, M.R., Gersten, S.M.: The optimal isoperimetric inequality for torus bundles over the circle. Q. J. Math. Oxford Ser. (2)
**47**(185), 1–23 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Bridson, M.R., Groves, D.: Free group automorphisms, train tracks and the beaded decomposition. arXiv:math/0507589v1, 37 pp
- 16.Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
- 17.Bridson, M.R., Pittet, Ch.: Isoperimetric inequalities for the fundamental groups of torus bundles over the circle. Geom. Dedicata
**49**(2), 203–219 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Button, J.O.: Free by cyclic groups and linear groups with restricted unipotent elements. Groups Complex. Cryptol.
**9**(2), 137–149 (2017)MathSciNetzbMATHGoogle Scholar - 19.Button, J.O.: Minimal dimension faithful linear representations of common finitely presented groups. arXiv:1610.03712, 39 pp
- 20.Button, J.: Tubular free by cyclic groups act freely on CAT(0) cube complexes. Canad. Math. Bull.
**60**(1), 54–62 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Dison, W.: An isoperimetric function for Bestvina–Brady groups. Bull. Lond. Math. Soc.
**40**(3), 384–394 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Dison, W., Riley, T.R.: Hydra groups. Comment. Math. Helv.
**88**(3), 507–540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Geoghegan, R.: Topological methods in group theory. Graduate Texts in Mathematics, 243. Springer, New York, 2008. xiv+473 ppGoogle Scholar
- 24.Gersten, S.M.: The automorphism group of a free group is not a \(\text{ CAT }(0)\) group. Proc. Am. Math. Soc.
**121**(4), 999–1002 (1994)MathSciNetzbMATHGoogle Scholar - 25.Hagen, M.F., Wise, D.T.: Cubulating hyperbolic free-by-cyclic groups: the irreducible case. Duke Math. J.
**165**(9), 1753–1813 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Hagen, M.F., Wise, D.T.: Cubulating hyperbolic free-by-cyclic groups: the general case. Geom. Funct. Anal.
**25**(1), 134–179 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Haglund, F., Wise, D.T.: Special cube complexes. Geom. Funct. Anal.
**17**(5), 1551–1620 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
- 29.Howie, J.: On the asphericity of ribbon disc complements. Trans. Am. Math. Soc.
**289**(1), 281–302 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Levitt, G.: Counting growth types of automorphisms of free groups. Geom. Funct. Anal.
**19**(4), 1119–1146 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Macura, N.: Detour functions and quasi-isometries. Q. J. Math.
**53**(2), 207–239 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Piggott, A.: Detecting the growth of free group automorphisms by their action on the homology of subgroups of finite index. arXiv:math/0409319v1, 59 pp
- 33.Sapir, M., Birget, J.-C., Rips, E.: Isoperimetric and isodiametric functions of groups. Ann. Math. (2)
**156**(2), 345–466 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Wise, D.T.: Cubular tubular groups. Amer. Math. Soc.
**366**(10), 5503–5521 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 35.Woodhouse, D.: Classifying finite dimensional cubulations of tubular groups. Mich. Math. J.
**65**(3), 511–532 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 36.Woodhouse, D.: Classifying virtually special tubular groups. Groups Geom. Dyn.
**12**(2), 679–702 (2018)MathSciNetCrossRefGoogle Scholar