The symmetry of intersection numbers in group theory.

*(English)*Zbl 0897.20029
Geom. Topol. 2, 11-29 (1998); correction ibid. 2, 333-335 (1998).

The geometric intersection number of two simple loops in a closed orientable surface is the least number of intersections of any loops isotopic to them. This is a case of the intersection number of two subgroups of the fundamental group of the surface, namely the two infinite cyclic subgroups carried by the loops. The author gives a definition of the intersection number of two subgroups of a finitely generated group in much greater (though not complete) generality, and proves that it is symmetric.

The author first defines the intersection number of certain almost invariant sets (a subset \(Z\) of a \(G\)-set is almost invariant if the symmetric difference \((Z-gZ)\cup(gZ-Z)\) is finite for all \(g\in G\)). For a subgroup \(H\) of \(G\), a subset \(X\) of \(G\) is called \(H\)-almost invariant if \(X\) is invariant under the left action of \(H\), and the quotient set \(H\backslash X\) is almost invariant under the right action of \(G\) on \(H\backslash G\). In addition, \(X\) is called a nontrivial \(H\)-almost invariant subset if \(H\backslash X\) and \(H\backslash(G-X)\) are both infinite. A \(\Lambda\)-almost invariant subset \(X\) is said to cross a \(\Sigma\)-almost invariant subset \(Y\) if each of the four sets \(X\cap Y\), \((G-X)\cap Y\), \(X\cap(G-Y)\), and \((G-X)\cap(G-Y)\) projects to an infinite subset of \(\Sigma\backslash G\). The author verifies that if \(X\) is a nontrivial \(\Lambda\)-almost invariant subset and \(Y\) is a nontrivial \(\Sigma\)-almost invariant subset, then \(X\) crosses \(Y\) if and only if \(Y\) crosses \(X\) (although this can fail if one of the sets is trivial). For nontrivial almost invariant subsets \(D\) and \(E\) of \(\Lambda\backslash G\) and \(\Sigma\backslash G\) respectively, with respective preimages \(X\) and \(Y\) in \(G\), the intersection number \(i(D,E)\) is defined to equal the number of double cosets \(\Sigma g\Lambda\) such that \(gX\) crosses \(Y\). When \(G\) if finitely generated, this number is finite. The bijection of \(G\) sending each element to its inverse shows that \(i\) is symmetric.

The intersection number of two subgroups \(\Lambda\) and \(\Sigma\) can be defined when the numbers of ends \(e(G,\Lambda)=e(G,\Sigma)=2\) (the number of ends of the pair \((G,\Lambda)\) is defined using any finite complex with fundamental group \(G\), it is then the number of ends of the covering space corresponding to \(\Lambda\)). The reason is that then there are, up to finite symmetric difference and taking complements, unique nontrivial almost invariant subsets \(D\) and \(E\) in \(\Lambda\backslash G\) and \(\Sigma\backslash G\), and one can define \(i(\Lambda,\Sigma)\) to be \(i(D,E)\). It can also be defined when \(G\) splits over \(\Lambda\) and \(\Sigma\), since there are then nontrivial almost invariant subsets determined by the splittings. However, in the latter case the intersection depends on the choice of splittings, not just on the subgroups.

The author closes with a nice interpretation of the intersection number of two subgroups, in the case where splittings of \(G\) over the subgroups \(\Lambda\) and \(\Sigma\) are given, and all three groups are finitely generated. If one takes the graph of groups determined by the splitting over \(\Lambda\), and takes the covering graph corresponding to \(\Sigma\), then the intersection number is the number of vertices of the smallest subgraph of this covering graph that carries its fundamental group \(\Sigma\).

The author first defines the intersection number of certain almost invariant sets (a subset \(Z\) of a \(G\)-set is almost invariant if the symmetric difference \((Z-gZ)\cup(gZ-Z)\) is finite for all \(g\in G\)). For a subgroup \(H\) of \(G\), a subset \(X\) of \(G\) is called \(H\)-almost invariant if \(X\) is invariant under the left action of \(H\), and the quotient set \(H\backslash X\) is almost invariant under the right action of \(G\) on \(H\backslash G\). In addition, \(X\) is called a nontrivial \(H\)-almost invariant subset if \(H\backslash X\) and \(H\backslash(G-X)\) are both infinite. A \(\Lambda\)-almost invariant subset \(X\) is said to cross a \(\Sigma\)-almost invariant subset \(Y\) if each of the four sets \(X\cap Y\), \((G-X)\cap Y\), \(X\cap(G-Y)\), and \((G-X)\cap(G-Y)\) projects to an infinite subset of \(\Sigma\backslash G\). The author verifies that if \(X\) is a nontrivial \(\Lambda\)-almost invariant subset and \(Y\) is a nontrivial \(\Sigma\)-almost invariant subset, then \(X\) crosses \(Y\) if and only if \(Y\) crosses \(X\) (although this can fail if one of the sets is trivial). For nontrivial almost invariant subsets \(D\) and \(E\) of \(\Lambda\backslash G\) and \(\Sigma\backslash G\) respectively, with respective preimages \(X\) and \(Y\) in \(G\), the intersection number \(i(D,E)\) is defined to equal the number of double cosets \(\Sigma g\Lambda\) such that \(gX\) crosses \(Y\). When \(G\) if finitely generated, this number is finite. The bijection of \(G\) sending each element to its inverse shows that \(i\) is symmetric.

The intersection number of two subgroups \(\Lambda\) and \(\Sigma\) can be defined when the numbers of ends \(e(G,\Lambda)=e(G,\Sigma)=2\) (the number of ends of the pair \((G,\Lambda)\) is defined using any finite complex with fundamental group \(G\), it is then the number of ends of the covering space corresponding to \(\Lambda\)). The reason is that then there are, up to finite symmetric difference and taking complements, unique nontrivial almost invariant subsets \(D\) and \(E\) in \(\Lambda\backslash G\) and \(\Sigma\backslash G\), and one can define \(i(\Lambda,\Sigma)\) to be \(i(D,E)\). It can also be defined when \(G\) splits over \(\Lambda\) and \(\Sigma\), since there are then nontrivial almost invariant subsets determined by the splittings. However, in the latter case the intersection depends on the choice of splittings, not just on the subgroups.

The author closes with a nice interpretation of the intersection number of two subgroups, in the case where splittings of \(G\) over the subgroups \(\Lambda\) and \(\Sigma\) are given, and all three groups are finitely generated. If one takes the graph of groups determined by the splitting over \(\Lambda\), and takes the covering graph corresponding to \(\Sigma\), then the intersection number is the number of vertices of the smallest subgraph of this covering graph that carries its fundamental group \(\Sigma\).

Reviewer: D.McCullough (Norman)

##### MSC:

20F65 | Geometric group theory |

57M07 | Topological methods in group theory |

20E07 | Subgroup theorems; subgroup growth |

##### Keywords:

fundamental groups; subgroups; intersection numbers; symmetry; JSJ splittings; almost invariant subgroups; graphs of groups; finitely generated groups; group actions; numbers of double cosets; numbers of ends; almost invariant sets**OpenURL**

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