Stallings theorem about ends of groups

In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group has more than one end if and only if the group admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group has more than one end if and only if admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.

The theorem was proved by John R. Stallings, first in the torsion-free case (1968)[1] and then in the general case (1971).[2]

Ends of graphs

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Let be a connected graph where the degree of every vertex is finite. One can view as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of are the ends of this topological space. A more explicit definition of the number of ends of a graph is presented below for completeness.

Let be a non-negative integer. The graph is said to satisfy if for every finite collection of edges of the graph has at most infinite connected components. By definition, if and if for every the statement is false. Thus if is the smallest nonnegative integer such that . If there does not exist an integer such that , put . The number is called the number of ends of .

Informally, is the number of "connected components at infinity" of . If , then for any finite set of edges of there exists a finite set of edges of with such that has exactly infinite connected components. If , then for any finite set of edges of and for any integer there exists a finite set of edges of with such that has at least infinite connected components.

Ends of groups

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Let be a finitely generated group. Let be a finite generating set of and let be the Cayley graph of with respect to . The number of ends of is defined as . A basic fact in the theory of ends of groups says that does not depend on the choice of a finite generating set of , so that is well-defined.

Basic facts and examples

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  • For a finitely generated group we have if and only if is finite.
  • For the infinite cyclic group we have
  • For the free abelian group of rank two we have
  • For a free group where we have .

Freudenthal-Hopf theorems

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Hans Freudenthal[3] and independently Heinz Hopf[4] established in the 1940s the following two facts:

Charles T. C. Wall proved in 1967 the following complementary fact:[5]

  • A group is virtually infinite cyclic if and only if it has a finite normal subgroup such that is either infinite cyclic or infinite dihedral.

Cuts and almost invariant sets

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Let be a finitely generated group, be a finite generating set of and let be the Cayley graph of with respect to . For a subset denote by the complement of in .

For a subset , the edge boundary or the co-boundary of consists of all (topological) edges of connecting a vertex from with a vertex from . Note that by definition .

An ordered pair is called a cut in if is finite. A cut is called essential if both the sets and are infinite.

A subset is called almost invariant if for every the symmetric difference between and is finite. It is easy to see that is a cut if and only if the sets and are almost invariant (equivalently, if and only if the set is almost invariant).

Cuts and ends

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A simple but important observation states:

if and only if there exists at least one essential cut in Γ.

Cuts and splittings over finite groups

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If where and are nontrivial finitely generated groups then the Cayley graph of has at least one essential cut and hence . Indeed, let and be finite generating sets for and accordingly so that is a finite generating set for and let be the Cayley graph of with respect to . Let consist of the trivial element and all the elements of whose normal form expressions for starts with a nontrivial element of . Thus consists of all elements of whose normal form expressions for starts with a nontrivial element of . It is not hard to see that is an essential cut in Γ so that .

A more precise version of this argument shows that for a finitely generated group :

  • If is a free product with amalgamation where is a finite group such that and then and are finitely generated and .
  • If is an HNN-extension where , are isomorphic finite subgroups of then is a finitely generated group and .

Stallings' theorem shows that the converse is also true.

Formal statement of Stallings' theorem

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Let be a finitely generated group.

Then if and only if one of the following holds:

  • The group admits a splitting as a free product with amalgamation where is a finite group such that and .
  • The group is an HNN extension where and , are isomorphic finite subgroups of .

In the language of Bass–Serre theory this result can be restated as follows: For a finitely generated group we have if and only if admits a nontrivial (that is, without a global fixed vertex) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.

For the case where is a torsion-free finitely generated group, Stallings' theorem implies that if and only if admits a proper free product decomposition with both and nontrivial.

Applications and generalizations

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  • Among the immediate applications of Stallings' theorem was a proof by Stallings[6] of a long-standing conjecture that every finitely generated group of cohomological dimension one is free and that every torsion-free virtually free group is free.
  • Stallings' theorem also implies that the property of having a nontrivial splitting over a finite subgroup is a quasi-isometry invariant of a finitely generated group since the number of ends of a finitely generated group is easily seen to be a quasi-isometry invariant. For this reason Stallings' theorem is considered to be one of the first results in geometric group theory.
  • Stallings' theorem was a starting point for Dunwoody's accessibility theory. A finitely generated group is said to be accessible if the process of iterated nontrivial splitting of over finite subgroups always terminates in a finite number of steps. In Bass–Serre theory terms that the number of edges in a reduced splitting of as the fundamental group of a graph of groups with finite edge groups is bounded by some constant depending on . Dunwoody proved[7] that every finitely presented group is accessible but that there do exist finitely generated groups that are not accessible.[8] Linnell[9] showed that if one bounds the size of finite subgroups over which the splittings are taken then every finitely generated group is accessible in this sense as well. These results in turn gave rise to other versions of accessibility such as Bestvina-Feighn accessibility[10] of finitely presented groups (where the so-called "small" splittings are considered), acylindrical accessibility,[11][12] strong accessibility,[13] and others.
  • Stallings' theorem is a key tool in proving that a finitely generated group is virtually free if and only if can be represented as the fundamental group of a finite graph of groups where all vertex and edge groups are finite (see, for example,[14]).
  • Using Dunwoody's accessibility result, Stallings' theorem about ends of groups and the fact that if is a finitely presented group with asymptotic dimension 1 then is virtually free[15] one can show [16] that for a finitely presented word-hyperbolic group the hyperbolic boundary of has topological dimension zero if and only if is virtually free.
  • Relative versions of Stallings' theorem and relative ends of finitely generated groups with respect to subgroups have also been considered. For a subgroup of a finitely generated group one defines the number of relative ends as the number of ends of the relative Cayley graph (the Schreier coset graph) of with respect to . The case where is called a semi-splitting of over . Early work on semi-splittings, inspired by Stallings' theorem, was done in the 1970s and 1980s by Scott,[17] Swarup,[18] and others.[19][20] The work of Sageev[21] and Gerasimov[22] in the 1990s showed that for a subgroup the condition corresponds to the group admitting an essential isometric action on a CAT(0)-cubing where a subgroup commensurable with stabilizes an essential "hyperplane" (a simplicial tree is an example of a CAT(0)-cubing where the hyperplanes are the midpoints of edges). In certain situations such a semi-splitting can be promoted to an actual algebraic splitting, typically over a subgroup commensurable with , such as for the case where is finite (Stallings' theorem). Another situation where an actual splitting can be obtained (modulo a few exceptions) is for semi-splittings over virtually polycyclic subgroups. Here the case of semi-splittings of word-hyperbolic groups over two-ended (virtually infinite cyclic) subgroups was treated by Scott-Swarup[23] and by Bowditch.[24] The case of semi-splittings of finitely generated groups with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of Dunwoody-Swenson.[25]
  • A number of new proofs of Stallings' theorem have been obtained by others after Stallings' original proof. Dunwoody gave a proof[26] based on the ideas of edge-cuts. Later Dunwoody also gave a proof of Stallings' theorem for finitely presented groups using the method of "tracks" on finite 2-complexes.[7] Niblo obtained a proof[27] of Stallings' theorem as a consequence of Sageev's CAT(0)-cubing relative version, where the CAT(0)-cubing is eventually promoted to being a tree. Niblo's paper also defines an abstract group-theoretic obstruction (which is a union of double cosets of in ) for obtaining an actual splitting from a semi-splitting. It is also possible to prove Stallings' theorem for finitely presented groups using Riemannian geometry techniques of minimal surfaces, where one first realizes a finitely presented group as the fundamental group of a compact -manifold (see, for example, a sketch of this argument in the survey article of Wall[28]). Gromov outlined a proof (see pp. 228–230 in [16]) where the minimal surfaces argument is replaced by an easier harmonic analysis argument and this approach was pushed further by Kapovich to cover the original case of finitely generated groups.[15][29]

See also

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Notes

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  1. ^ John R. Stallings. On torsion-free groups with infinitely many ends. Annals of Mathematics (2), vol. 88 (1968), pp. 312–334
  2. ^ John Stallings. Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.-London, 1971.
  3. ^ H. Freudenthal. Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv. 17, (1945). 1-38.
  4. ^ H. Hopf. Enden offener Räume und unendliche diskontinuierliche Gruppen. Comment. Math. Helv. 16, (1944). 81-100
  5. ^ Lemma 4.1 in C. T. C. Wall, Poincaré Complexes: I. Annals of Mathematics, Second Series, Vol. 86, No. 2 (Sep., 1967), pp. 213-245
  6. ^ John R. Stallings. Groups of dimension 1 are locally free. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364
  7. ^ a b M. J. Dunwoody. The accessibility of finitely presented groups. Inventiones Mathematicae, vol. 81 (1985), no. 3, pp. 449-457
  8. ^ M. J. Dunwoody. An inaccessible group. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 75–78, London Mathematical Society Lecture Note Series, vol. 181, Cambridge University Press, Cambridge, 1993; ISBN 0-521-43529-3
  9. ^ Linnell, P. A. (1983). "On accessibility of groups". Journal of Pure and Applied Algebra. 30 (1): 39–46. doi:10.1016/0022-4049(83)90037-3. MR 0716233.
  10. ^ M. Bestvina and M. Feighn. Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no. 3, pp. 449–469
  11. ^ Z. Sela. Acylindrical accessibility for groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 527–565
  12. ^ T. Delzant. Sur l'accessibilité acylindrique des groupes de présentation finie. Archived 2011-06-05 at the Wayback Machine Université de Grenoble. Annales de l'Institut Fourier, vol. 49 (1999), no. 4, pp. 1215–1224
  13. ^ Delzant, Thomas; Potyagailo, Leonid (2001). "Accessibilité hiérarchique des groupes de présentation finie". Topology. 40 (3): 617–629. doi:10.1016/S0040-9383(99)00078-6. MR 1838998.
  14. ^ H. Bass. Covering theory for graphs of groups. Journal of Pure and Applied Algebra, vol. 89 (1993), no. 1-2, pp. 3–47
  15. ^ a b Gentimis Thanos, Asymptotic dimension of finitely presented groups, http://www.ams.org/journals/proc/2008-136-12/S0002-9939-08-08973-9/home.html
  16. ^ a b M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263
  17. ^ Scott, Peter (1977–1978). "Ends of pairs of groups". Journal of Pure and Applied Algebra. 11 (1–3): 179–198. doi:10.1016/0022-4049(77)90051-2. MR 0487104.
  18. ^ Swarup, G. Ananda (1977–1978). "Relative version of a theorem of Stallings". Journal of Pure and Applied Algebra. 11 (1–3): 75–82. doi:10.1016/0022-4049(77)90042-1. MR 0466326.
  19. ^ H. Müller. Decomposition theorems for group pairs. Mathematische Zeitschrift, vol. 176 (1981), no. 2, pp. 223–246
  20. ^ Kropholler, P. H.; Roller, M. A. (1989). "Relative ends and duality groups". Journal of Pure and Applied Algebra. 61 (2): 197–210. doi:10.1016/0022-4049(89)90014-5. MR 1025923.
  21. ^ Michah Sageev. Ends of group pairs and non-positively curved cube complexes. Proceedings of the London Mathematical Society (3), vol. 71 (1995), no. 3, pp. 585–617
  22. ^ V. N. Gerasimov. Semi-splittings of groups and actions on cubings. (in Russian) Algebra, geometry, analysis and mathematical physics (Novosibirsk, 1996), pp. 91–109, 190, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997
  23. ^ G. P. Scott, and G. A. Swarup. An algebraic annulus theorem. Archived 2007-07-15 at the Wayback Machine Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506
  24. ^ B. H. Bowditch. Cut points and canonical splittings of hyperbolic groups. Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186
  25. ^ M. J. Dunwoody, and E. L. Swenson. The algebraic torus theorem. Inventiones Mathematicae, vol. 140 (2000), no. 3, pp. 605–637
  26. ^ M. J. Dunwoody. Cutting up graphs. Combinatorica, vol. 2 (1982), no. 1, pp. 15–23
  27. ^ Graham A. Niblo. A geometric proof of Stallings' theorem on groups with more than one end. Geometriae Dedicata, vol. 105 (2004), pp. 61–76
  28. ^ C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5–101
  29. ^ M. Kapovich. Energy of harmonic functions and Gromov's proof of Stallings' theorem, preprint, 2007, arXiv:0707.4231