×

Homological conditions for graphical splittings of antisocial graph groups. (English) Zbl 1020.20028

Given a finite graph \(X\) with vertex set \(V\) and edge set \(E\), the ‘graph group’ \(G(X)\) is generated by the elements of \(V\), with defining relations saying that two elements of \(V\) commute if and only if they are joined by an edge of \(X\).
By J. R. Stallings’s ends theorem [Ann. Math. (2) 88, 312-334 (1968; Zbl 0238.20036)], a group has more than one end if and only if it splits as either a free product with finite amalgamated subgroups, or as an HNN extension with finite associated subgroups. Since graph groups are torsion-free, this means that the only ones with more than one end are \(\mathbb{Z}\) and those that are non-trivial free products (i.e., the underlying graph is disconnected).
A graph group \(G(X)\) is said to be ‘graphically \(n\)-decomposable’ if there is a subgraph \(Y\subseteq X\) such that \(Y\) has a maximal complete subgraph with \(n\) vertices, and \(G(X)\) splits as either (i) an HNN-extension over \(G(Y)\), or (ii) a free product with amalgamation \(G(X_1)*_{G(Y)}G(X_2)\), where \(G(Y)\) is a maximal Abelian subgroup of neither \(G(X_1)\) nor \(G(X_2)\). (Note that for \(n=0\), this means \(Y=\emptyset\), and so \(G(X)\) is either (i) an HNN extension over the trivial group (i.e., \(G(X)=\mathbb{Z}\)) or (ii) a free product with nontrivial factors and trivial amalgamated subgroup – that is, \(G(X)\) has more than one end.)
There is a well-known cube complex \(\text{Cube}(X)\) on which \(G(X)\) acts; define \(b_n(X)\) to be the \(n\)-th Betti number of the one-sphere of \(\text{Cube}(X)\). Then \(b_0(X)\neq 0\) if and only if \(X\) either consists of a single vertex, or if \(X\) is disconnected – in other words, \(G(X)\) is graphically \(0\)-decomposable if and only if \(b_0(X)\neq 0\).
The author proves that for certain graphs (the “antisocial” ones), the graph group \(G(X)\) is graphically \(n\)-decomposable iff \(b_n(X)\neq 0\).

MSC:

20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations

Citations:

Zbl 0238.20036
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bestvina, M.; Brady, N., Morse theory and finiteness properties of groups, Invent. Math., 129, 3, 445-470 (1997) · Zbl 0888.20021
[2] Charney, R., Artin groups of finite type are biautomatic, Math. Ann., 292, 4, 671-683 (1992) · Zbl 0736.57001
[3] Dicks, W.; Dunwoody, M. J., Groups Acting on Graphs. Groups Acting on Graphs, Cambridge Stud. in Adv. Math., 17 (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0665.20001
[4] Harary, F., Graph Theory (1969), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0797.05064
[5] Hermiller, S.; Meier, J., Algorithms and geometry for graph products of groups, J. Algebra, 171, 230-257 (1995) · Zbl 0831.20032
[6] Stallings, J. R., On torsion-free groups with infinitely many ends, Ann. of Math., 88, 312-334 (1968) · Zbl 0238.20036
[7] Stallings, J. R., Group Theory and Three Dimensional Manifolds (1971), Yale University Press: Yale University Press New Haven, CT · Zbl 0241.57001
[8] Scott, P.; Wall, C. T.C., Topological methods in group theory, (Homological Group Theory. Homological Group Theory, London Math. Soc. Lecture Note Ser., 36 (1979), Cambridge University Press: Cambridge University Press Cambridge), 137-203
[9] van Wyk, L., Graph groups are biautomatic, J. Pure Appl. Algebra, 94, 341-352 (1994) · Zbl 0812.20018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.