Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T15:19:29.267Z Has data issue: false hasContentIssue false
  • English
  • Français

Revisiting Leighton’s theorem with the Haar measure

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  13 January 2020

DANIEL J. WOODHOUSE
DANIEL J. WOODHOUSE*
Affiliation:
Postal address: Mathematics Department Technion – Israel Institute of Technology HAIFA - 32000, ISRAEL. e-mail: daniel.woodhouse@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

MSC classification

Primary: 57M10: Covering spaces
Secondary: 20E08: Groups acting on trees 20E05: Free nonabelian groups 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 615 - 623
Copyright
© Cambridge Philosophical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Present address: Mathematical Institute University of Oxford Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road, Oxford, OX2 6GG.

†

Supported by the Israel Science Foundation (grant 1026/15).

References

REFERENCES

Angluin, D. and Gardiner, A.. Finite common coverings of pairs of regular graphs. J. Combin. Theory Ser. B, 30(2) (1981), 184–187.CrossRefGoogle Scholar
Bass, H.. Covering theory for graphs of groups. J. Pure Appl. Algebra 89(1-2) (1993), 3–47.CrossRefGoogle Scholar
Bass, H. and Kulkarni, R.. Uniform tree lattices. J. Amer. Math. Soc. 3(4) (1990), 843–902.CrossRefGoogle Scholar
Behrstock, J. A. and Neumann, W. D.. Quasi-isometric classification of non-geometric 3-manifold groups. J. Reine Angew. Math. 669 (2012), 101–120.Google Scholar
Biswas, K. and Mj, M.. Pattern rigidity in hyperbolic spaces: duality and PD subgroups. Groups Geom. Dyn. 6(1) (2012), 97–123.CrossRefGoogle Scholar
Cashen, C. H. and Macura, N.. Line patterns in free groups. Geom. Topol. 15(3) (2011), 1419–1475.CrossRefGoogle Scholar
Cashen, C. H. and Martin, A.. Quasi-isometries between groups with two-ended splittings. Math. Proc. Camb. Phil. Soc. 162(2) (2017), 249–291.CrossRefGoogle Scholar
Hagen, M. F. and Touikan, N. W. M.. Panel collapse and its applications. To appear in Groups, Geometry and Dynamics, https://arxiv.org/abs/1712.06553.Google Scholar
Leighton, F. T.. Finite common coverings of graphs. J. Combin. Theory Ser. B 33(3) (1982), 231–238.CrossRefGoogle Scholar
Levitt, G.. Generalised Baumslag–Solitar groups: rank and finite index subgroups. Ann. Inst. Fourier (Grenoble), 65(2) (2015), 725–762.CrossRefGoogle Scholar
Mj, M.. Pattern rigidity and the Hilbert-Smith conjecture. Geom. Topol. 16(2) (2012), 1205–1246.CrossRefGoogle Scholar
Mosher, L., Sageev, M. and Whyte, K.. Quasi-actions on trees II: Finite depth Bass–Serre trees. Mem. Amer. Math. Soc. 214(1008) (2011), vi+105.Google Scholar
Neumann, W. D.. On Leighton’s graph covering theorem. Groups Geom. Dyn. 4(4) (2010), 863–872.CrossRefGoogle Scholar
Papasoglu, P.. Quasi-isometry invariance of group splittings. Ann. of Math. (2) 161(2) (2005), 759–830.CrossRefGoogle Scholar
Schwartz, R. E.. Symmetric patterns of geodesics and automorphisms of surface groups. Invent. Math. 128(1) (1997), 177–199.CrossRefGoogle Scholar
Taam, A. and Touikan, N. W. M.. On the quasi-isometric rigidity of graphs of surface groups. Preprint, https://arxiv.org/abs/1904.10482.Google Scholar
Whitehead, J. H. C.. On equivalent sets of elements in a free group. Ann. of Math. (2) 37(4) (1936), 782–800.CrossRefGoogle Scholar