Ends of group pairs and non-positively curved cube complexes.

*(English)*Zbl 0861.20041A group is called splittable if it is an HNN extension or a nontrivial free product with amalgamation. In geometric group theory, this type of structure is seen in terms of an action of the group on a topological object. A central theorem of the Bass-Serre theory of group actions on trees is that \(G\) is splittable if and only if \(G\) acts (simplicially) on a tree without a global fixed point. Splittability is related to the theory of ends of groups. The number of ends of a finitely generated group \(G\) is defined to be the number \(e(G)\) of topological ends of its Cayley complex, and is independent of the choice of generating set used to define the complex. The number of ends is 0 for finite groups and 2 for groups which contain an infinite cyclic subgroup of finite index. For all other groups it is either 1 or \(\infty\). A celebrated theorem of J. Stallings characterizes groups with \(e(G)>1\) as those which are splittable over a finite subgroup.

For a pair \((G,H)\) with \(H\) a subgroup of \(G\), one can similarly define \(e(G,H)\) to be the number of topological ends of the quotient of the Cayley graph of \(G\) by the action of \(H\). Here \(e(G,H)\) can be any nonnegative integer or \(\infty\). This notion was introduced by C. H. Houghton [J. Aust. Math. Soc. 17, 274-284 (1974; Zbl 0289.22005)], and substantially developed by P. Scott [J. Pure Appl. Algebra 11, 179-198 (1977; Zbl 0368.20021)]. One defines \((G,H)\) to be multiended if \(e(G,H)>1\), and calls \(G\) semisplittable if it is multiended with respect to some subgroup. Splittable groups are semisplittable, but Scott gave examples of semisplittable groups which are not splittable. He showed, however, that if \(G\) is \(H\)-residually finite, then \(G\) contains a finite-index subgroup which splits over \(H\).

In this paper the author develops a theory of actions of group pairs in the spirit of the Bass-Serre theory of actions on trees. In place of trees one has simply connected cube complexes of nonpositive curvature, called cubings. Each \(n\)-cell has the structure of a cube, each of whose faces is identified with an \((n-1)\)-dimensional cell. The group acts cellularly, but the condition of essentiality is more complicated than simply failing to have a global fixed point. There is a concept of codimension-1 hyperplane, which in the case of trees is just the midpoint of an edge. These separate the cubing \(X\) into two components. Then, each vertex \(v\) determines a partition of \(G\) into two subsets according to which component contains the image of \(v\) under the group element. The action is called essential (with respect to the hyperplane \(J\)) if the corresponding partition of \(G\) is such that both parts contain infinitely many right cosets of the stabilizer of \(J\). The main theorem is that a finitely generated group \(G\) is semisplittable if and only if \(G\) acts essentially on a cubing. When the cubing is finite-dimensional, the essentiality condition simplifies to the property that there exists an unbounded orbit.

The author applies the theory to prove that if \(M\) is a closed orientable irreducible 3-manifold and \(\pi_1(M)\) acts on a cubing with unbounded orbit, then \(M\) contains an immersed incompressible surface. On the other hand, if \(M\) contains an immersed incompressible surface, the theory produces an essential action of \(\pi_1(M)\) on a cubing. When \(M\) contains an immersed incompressible surface satisfying the \(k\)-plane property of J. Hass and P. Scott [Topology 31, No. 3, 493-517 (1992; Zbl 0771.57007)], an essential action of \(\pi_1(M)\) on a \(k\)-dimensional cubing can be constructed.

The author develops the theory carefully and clearly, providing a firm foundation for its further development.

For a pair \((G,H)\) with \(H\) a subgroup of \(G\), one can similarly define \(e(G,H)\) to be the number of topological ends of the quotient of the Cayley graph of \(G\) by the action of \(H\). Here \(e(G,H)\) can be any nonnegative integer or \(\infty\). This notion was introduced by C. H. Houghton [J. Aust. Math. Soc. 17, 274-284 (1974; Zbl 0289.22005)], and substantially developed by P. Scott [J. Pure Appl. Algebra 11, 179-198 (1977; Zbl 0368.20021)]. One defines \((G,H)\) to be multiended if \(e(G,H)>1\), and calls \(G\) semisplittable if it is multiended with respect to some subgroup. Splittable groups are semisplittable, but Scott gave examples of semisplittable groups which are not splittable. He showed, however, that if \(G\) is \(H\)-residually finite, then \(G\) contains a finite-index subgroup which splits over \(H\).

In this paper the author develops a theory of actions of group pairs in the spirit of the Bass-Serre theory of actions on trees. In place of trees one has simply connected cube complexes of nonpositive curvature, called cubings. Each \(n\)-cell has the structure of a cube, each of whose faces is identified with an \((n-1)\)-dimensional cell. The group acts cellularly, but the condition of essentiality is more complicated than simply failing to have a global fixed point. There is a concept of codimension-1 hyperplane, which in the case of trees is just the midpoint of an edge. These separate the cubing \(X\) into two components. Then, each vertex \(v\) determines a partition of \(G\) into two subsets according to which component contains the image of \(v\) under the group element. The action is called essential (with respect to the hyperplane \(J\)) if the corresponding partition of \(G\) is such that both parts contain infinitely many right cosets of the stabilizer of \(J\). The main theorem is that a finitely generated group \(G\) is semisplittable if and only if \(G\) acts essentially on a cubing. When the cubing is finite-dimensional, the essentiality condition simplifies to the property that there exists an unbounded orbit.

The author applies the theory to prove that if \(M\) is a closed orientable irreducible 3-manifold and \(\pi_1(M)\) acts on a cubing with unbounded orbit, then \(M\) contains an immersed incompressible surface. On the other hand, if \(M\) contains an immersed incompressible surface, the theory produces an essential action of \(\pi_1(M)\) on a cubing. When \(M\) contains an immersed incompressible surface satisfying the \(k\)-plane property of J. Hass and P. Scott [Topology 31, No. 3, 493-517 (1992; Zbl 0771.57007)], an essential action of \(\pi_1(M)\) on a \(k\)-dimensional cubing can be constructed.

The author develops the theory carefully and clearly, providing a firm foundation for its further development.

Reviewer: D.McCullough (Norman)

##### MSC:

20F65 | Geometric group theory |

20E08 | Groups acting on trees |

57M07 | Topological methods in group theory |

57M60 | Group actions on manifolds and cell complexes in low dimensions |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |