Morse theory and finiteness properties of groups.

*(English)*Zbl 0888.20021J. Stallings gave an elegant example of a group which is finitely generated but not finitely presented. Let \(G=F_2\times F_2\) be the direct product of two free groups of rank 2. The example is the kernel of the homomorphism from \(G\) to \(\mathbb{Z}\) that sends each generator to 1. In this paper, the authors give far-reaching generalizations of this example, in a geometric framework that allows them to resolve some longstanding homological finiteness questions. Moreover, they show that at least one of two well-known conjectures – the Eilenberg-Ganea Conjecture and the Whitehead Asphericity Conjecture – must fail.

A simplicial complex \(L\) is called a flag complex if every finite collection of vertices of \(L\) which are pairwise adjacent spans a simplex in \(L\). This places no topological restriction on \(L\), since the first barycentric subdivision of any simplicial complex is a flag complex. The right-angled Artin group \(G_L\) associated to a finite flag complex \(L\) is defined by taking one generator for each vertex of \(L\), and declaring that for each edge the generators corresponding to its endpoints commute. For example, \(F_2\times F_2\) is associated to the flag complex which is the boundary of the square. A natural homomorphism \(G_L\to\mathbb{Z}\) is defined by sending each generator to 1, and \(H_L\) denotes its kernel. The geometric machinery of the paper yields an Eilenberg-MacLane space \(Q_L\) for \(G_L\) which is a compact, piecewise euclidean cubical complex of nonpositive curvature, and a continuous map \(Q_L\to\mathbb{Z}\) which induces the homomorphism \(G_L\to\mathbb{Z}\) on fundamental groups. This map lifts to a “Morse function” from the universal covering of \(Q_L\) to \(\mathbb{R}\). Then, \(H_L\) acts cocompactly on the level sets of this function. An extensive analysis of this setup using geometric and topological methods yields the main theorem (where homology is taken with coefficients in a ring \(R\) with \(0\neq 1\)): (1) \(H_L\in\text{FP}_{n+1}(R)\) (i. e. \(R\) has a resolution \(P_{n+1}\to P_n\to\cdots\to P_0\to R\to 0\) by finitely generated projective \(RH\)-modules) if and only if \(L\) is homologically \(n\)-connected (2) \(H_L\in\text{FP}(R)\) (i. e. \(R\) has a finite length resolution \(0\to P_n\to\cdots\to P_0\to R\to 0\) by finitely generated projective \(RH\)-modules) if and only if \(L\) is acyclic, and (3) \(H_L\) is finitely presented if and only if \(L\) is simply connected. Choosing \(L\) to be \(\mathbb{Z}\)-acyclic but not simply connected yields an \(H_L\) which is in \(\text{FP}(\mathbb{Z})\) but is not finitely presented, resolving a longstanding question. Similarly, one obtains examples of groups which are in \(\text{FP}_n(R)\) but not \(\text{FP}_{n+1}(R)\) and so on. In the case \(R=\mathbb{Z}\) these replicate results of Stallings, Bieri, Stuhler, and Abels-Brown.

The Eilenberg-Ganea Conjecture is that a group of cohomological dimension 2 must be the fundamental group of a 2-dimensional aspherical complex, and the Whitehead Asphericity Conjecture is that any connected subcomplex of an aspherical 2-complex is aspherical. Taking \(L\) to be a flag triangulation of a spine of the PoincarĂ© homology sphere, a nontrivial application of the geometric machinery shows that if \(H_L\) acts freely, faithfully, properly, and cellularly on a contractible 2-complex \(Y\), then \(Y\) has a subcomplex with nontrivial \(\pi_2\). Thus either no such \(Y\) exists, and \(H_L\) is a counterexample to the Eilenberg-Ganea Conjecture, or a counterexample to the Whitehead Asphericity Conjecture is obtained.

In a remark added in proof, the authors note that J. Meier, H. Meinert, and L. Van Wyk have generalized the main theorem to include the kernels of all (including irrational) characters \(G_L\to\mathbb{R}\).

A simplicial complex \(L\) is called a flag complex if every finite collection of vertices of \(L\) which are pairwise adjacent spans a simplex in \(L\). This places no topological restriction on \(L\), since the first barycentric subdivision of any simplicial complex is a flag complex. The right-angled Artin group \(G_L\) associated to a finite flag complex \(L\) is defined by taking one generator for each vertex of \(L\), and declaring that for each edge the generators corresponding to its endpoints commute. For example, \(F_2\times F_2\) is associated to the flag complex which is the boundary of the square. A natural homomorphism \(G_L\to\mathbb{Z}\) is defined by sending each generator to 1, and \(H_L\) denotes its kernel. The geometric machinery of the paper yields an Eilenberg-MacLane space \(Q_L\) for \(G_L\) which is a compact, piecewise euclidean cubical complex of nonpositive curvature, and a continuous map \(Q_L\to\mathbb{Z}\) which induces the homomorphism \(G_L\to\mathbb{Z}\) on fundamental groups. This map lifts to a “Morse function” from the universal covering of \(Q_L\) to \(\mathbb{R}\). Then, \(H_L\) acts cocompactly on the level sets of this function. An extensive analysis of this setup using geometric and topological methods yields the main theorem (where homology is taken with coefficients in a ring \(R\) with \(0\neq 1\)): (1) \(H_L\in\text{FP}_{n+1}(R)\) (i. e. \(R\) has a resolution \(P_{n+1}\to P_n\to\cdots\to P_0\to R\to 0\) by finitely generated projective \(RH\)-modules) if and only if \(L\) is homologically \(n\)-connected (2) \(H_L\in\text{FP}(R)\) (i. e. \(R\) has a finite length resolution \(0\to P_n\to\cdots\to P_0\to R\to 0\) by finitely generated projective \(RH\)-modules) if and only if \(L\) is acyclic, and (3) \(H_L\) is finitely presented if and only if \(L\) is simply connected. Choosing \(L\) to be \(\mathbb{Z}\)-acyclic but not simply connected yields an \(H_L\) which is in \(\text{FP}(\mathbb{Z})\) but is not finitely presented, resolving a longstanding question. Similarly, one obtains examples of groups which are in \(\text{FP}_n(R)\) but not \(\text{FP}_{n+1}(R)\) and so on. In the case \(R=\mathbb{Z}\) these replicate results of Stallings, Bieri, Stuhler, and Abels-Brown.

The Eilenberg-Ganea Conjecture is that a group of cohomological dimension 2 must be the fundamental group of a 2-dimensional aspherical complex, and the Whitehead Asphericity Conjecture is that any connected subcomplex of an aspherical 2-complex is aspherical. Taking \(L\) to be a flag triangulation of a spine of the PoincarĂ© homology sphere, a nontrivial application of the geometric machinery shows that if \(H_L\) acts freely, faithfully, properly, and cellularly on a contractible 2-complex \(Y\), then \(Y\) has a subcomplex with nontrivial \(\pi_2\). Thus either no such \(Y\) exists, and \(H_L\) is a counterexample to the Eilenberg-Ganea Conjecture, or a counterexample to the Whitehead Asphericity Conjecture is obtained.

In a remark added in proof, the authors note that J. Meier, H. Meinert, and L. Van Wyk have generalized the main theorem to include the kernels of all (including irrational) characters \(G_L\to\mathbb{R}\).

Reviewer: D.McCullough (Norman)

##### MSC:

20F65 | Geometric group theory |

57M07 | Topological methods in group theory |

20F05 | Generators, relations, and presentations of groups |