## Algebraic & Geometric Topology

### Splittings of non-finitely generated groups

Robin M Lassonde

#### Abstract

In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersection numbers of splittings of finitely generated groups (as developed by Scott, Swarup, Niblo and Sageev), and rework it to remove finite generation assumptions. I show that when working with splittings, instead of using the Cayley graph, one can use Bass–Serre trees.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 1 (2012), 511-563.

Dates
Revised: 14 October 2011
Accepted: 12 December 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715349

Digital Object Identifier
doi:10.2140/agt.2012.12.511

Mathematical Reviews number (MathSciNet)
MR2916286

Zentralblatt MATH identifier
1282.20047

Keywords
splitting intersection number

#### Citation

Lassonde, Robin M. Splittings of non-finitely generated groups. Algebr. Geom. Topol. 12 (2012), no. 1, 511--563. doi:10.2140/agt.2012.12.511. https://projecteuclid.org/euclid.agt/1513715349

#### References

• B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145–186
• D E Cohen, Ends and free products of groups, Math. Z. 114 (1970) 9–18
• M J Dunwoody, Accessibility and groups of cohomological dimension one, Proc. London Math. Soc. 38 (1979) 193–215
• M J Dunwoody, M E Sageev, JSJ–splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999) 25–44
• M J Dunwoody, E L Swenson, The algebraic torus theorem, Invent. Math. 140 (2000) 605–637
• H Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931) 692–713
• K Fujiwara, P Papasoglu, JSJ–decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal. 16 (2006) 70–125
• V Guirardel, Cœ ur et nombre d'intersection pour les actions de groupes sur les arbres, Ann. Sci. École Norm. Sup. 38 (2005) 847–888
• V Guirardel, G Levitt, JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space
• V Guirardel, G Levitt, JSJ decompositions: definitions, existence, uniqueness. II: Compatibility and acylindricity
• G Higman, B H Neumann, H Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949) 247–254
• H Hopf, Enden offener Räume und unendliche diskontinuierliche Gruppen, Comment. Math. Helv. 16 (1944) 81–100
• C H Houghton, Ends of locally compact groups and their coset spaces, J. Austral. Math. Soc. 17 (1974) 274–284 Collection of articles dedicated to the memory of Hanna Neumann, VII
• W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 21, no. 220, Amer. Math. Soc. (1979)
• K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Math. 761, Springer, Berlin (1979)
• P H Kropholler, An analogue of the torus decomposition theorem for certain Poincaré duality groups, Proc. London Math. Soc. 60 (1990) 503–529
• H Müller, Decomposition theorems for group pairs, Math. Z. 176 (1981) 223–246
• G A Niblo, A geometric proof of Stallings' theorem on groups with more than one end, Geom. Dedicata 105 (2004) 61–76
• G A Niblo, M Sageev, P Scott, G A Swarup, Minimal cubings, Internat. J. Algebra Comput. 15 (2005) 343–366
• E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997) 53–109
• M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585–617
• O Schreier, Die Untergruppen der freien Gruppen, Abh. Mat Sem. Univ. Hamburg 5 (1927) 161–183
• P Scott, The symmetry of intersection numbers in group theory, Geom. Topol. 2 (1998) 11–29
• P Scott, G A Swarup, Splittings of groups and intersection numbers, Geom. Topol. 4 (2000) 179–218
• P Scott, G A Swarup, Regular neighbourhoods and canonical decompositions for groups, Astérisque 289, Soc. Math. France (2003)
• P Scott, G A Swarup, Errata for “Regular neighbourhoods and canonical decompositions for groups” [Electron. Res. Announc. Amer. Math. Soc. 8 (2002) 20–28 \xoxMR1928498], preprint (2006) Available at \setbox0\makeatletter\@url http://www.math.lsa.umich.edu/~pscott/preprints.html {\unhbox0
• P Scott, T Wall, Topological methods in group theory, from: “Homological group theory (Proc. Sympos., Durham, 1977)”, (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137–203
• Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups II, Geom. Funct. Anal. 7 (1997) 561–593
• J-P Serre, Arbres, amalgames, ${\rm SL}\sb{2}$, Astérisque 46, Soc. Math. France (1977)
• J-P Serre, Trees, Springer, Berlin (1980) Translated from the French by J Stillwell
• E Specker, Endenverbände von Räumen und Gruppen, Math. Ann. 122 (1950) 167–174
• J R Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. 88 (1968) 312–334
• J R Stallings, Group theory and $3$–manifolds, from: “Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2”, Gauthier-Villars, Paris (1971) 165–167
• R G Swan, Groups of cohomological dimension one, J. Algebra 12 (1969) 585–610
• F Waldhausen, On the determination of some bounded $3$–manifolds by their fundamental groups alone, from: “Proc. Sympos. on Topology and its Applications, Beograd”, Herceg-Novi, Yugoslavia (1969) 331–332