Poorly connected groups

Hume, David; Mackay, John

doi:10.1090/proc/15128

Poorly connected groups
HTML articles powered by AMS MathViewer

by David Hume and John M. Mackay PDF

Proc. Amer. Math. Soc. 148 (2020), 4653-4664

Abstract:

We investigate groups whose Cayley graphs have poorly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini–Schramm–Timár if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag–Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.

References

Martin R. Bridson and Daniel Groves, The quadratic isoperimetric inequality for mapping tori of free group automorphisms, Mem. Amer. Math. Soc. 203 (2010), no. 955, xii+152. MR 2590896, DOI 10.1090/S0065-9266-09-00578-X
Itai Benjamini, Oded Schramm, and Ádám Timár, On the separation profile of infinite graphs, Groups Geom. Dyn. 6 (2012), no. 4, 639–658. MR 2996405, DOI 10.4171/GGD/168
Matthew Cordes and David Hume, Stability and the Morse boundary, J. Lond. Math. Soc. (2) 95 (2017), no. 3, 963–988. MR 3664526, DOI 10.1112/jlms.12042
Cornelia Druţu and Michael Kapovich, Geometric group theory, American Mathematical Society Colloquium Publications, vol. 63, American Mathematical Society, Providence, RI, 2018. With an appendix by Bogdan Nica. MR 3753580, DOI 10.1090/coll/063
Volker Diekert and Armin Weiß, Context-free groups and Bass-Serre theory, Algorithmic and geometric topics around free groups and automorphisms, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2017, pp. 43–110. MR 3729161
M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI 10.1007/BF01388581
David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694
Nima Hoda, Shortcut Graphs and Groups, Preprint, arXiv:1811.05036 (2018).
David Hume, John M. Mackay, and Romain Tessera, Poincaré profiles of groups and spaces, Preprint, arXiv:1707.02151 (2017); Rev. Math. Iberoam. (to appear).
David Hume, John M. Mackay, and Romain Tessera, Poincaré profiles of connected unimodular Lie groups, Provisional title. In preparation, 2019.
David Hume, Infinitely presented group where every finite sub-presentation is virtually free, MathOverflow. https://mathoverflow.net/q/323660 (version: 2019-02-21).
David Hume, A continuum of expanders, Fund. Math. 238 (2017), no. 2, 143–152. MR 3640615, DOI 10.4064/fm101-11-2016
Dietrich Kuske and Markus Lohrey, Logical aspects of Cayley-graphs: the group case, Ann. Pure Appl. Logic 131 (2005), no. 1-3, 263–286. MR 2097229, DOI 10.1016/j.apal.2004.06.002
John M. Mackay and Alessandro Sisto, Quasi-hyperbolic planes in relatively hyperbolic groups, Preprint, arXiv:1111.2499, 2011. To appear in Ann. Sci. Fenn.
Jason Fox Manning, Geometry of pseudocharacters, Geom. Topol. 9 (2005), 1147–1185. MR 2174263, DOI 10.2140/gt.2005.9.1147
Alexander Yu. Ol′shanskii, Denis V. Osin, and Mark V. Sapir, Lacunary hyperbolic groups, Geom. Topol. 13 (2009), no. 4, 2051–2140. With an appendix by Michael Kapovich and Bruce Kleiner. MR 2507115, DOI 10.2140/gt.2009.13.2051
P. Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent. Math. 121 (1995), no. 2, 323–334. MR 1346209, DOI 10.1007/BF01884301