Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes. (English) Zbl 1400.20023
I. Kapovich and A. Myasnikov [J. Algebra 248, No. 2, 608–668 (2002; Zbl 1001.20015)] described techniques to study subgroups of free groups via graphs and their foldings. The author of the present paper generalizes these techniques to study finite index subgroups of finitely generated (though not necessarily finitely presented) groups.
For each such group \(G\) with presentation \(\langle\, X \mid R\,\rangle\) and each subgroup \(H\) of \(G\) of finite index \(n\), she associates a connected graph with \(n\) vertices whose edges are labeled by elements of \(X\) and whose closed paths reflect the relators \(R\). This subgroup graph, which is revealed to be the Schreier coset graph of \(H\) with respect to \(X\), carries much essential information about \(H\), and is unique up to isomorphism in this category. Furthermore, she identifies \(H\) as the language subgroup (in the sense of graph automata) of the subgroup graph. A first collection of applications describe how various properties of subgroups – for example, the conjugacy of subgroups and the normality of a subgroup – are reflected in their subgroup graphs.
Note that the main results on subgroup graphs apply to finite groups and their subgroups. The author shows how subgroup graphs identify Sylow (and, more generally, Hall) subgroups of a finite group, and uses these tools to prove an easy (and known) divisibility property involving a finite group with a malnormal subgroup (i.e. a group with a Frobenius kernel).
The second application relates to the poset \(P_{\mathrm{fi}}(G)\) of cosets of proper finite index subgroups of \(G\) under inclusion and its associated simplicial complex of chains \(\Delta(P_{\mathrm{fi}}(G))\). When \(G\) is finite, \(\Delta(P_{\mathrm{fi}}(G))\) is known not to be contractible. The author exhibits sufficient conditions for contractibility of \(\Delta(P_{\mathrm{fi}}(G))\) for a finitely generated group in terms of the existence of a family of subgroup graphs of selected finite index subgroups of \(G\). She then verifies this condition for a number of families of infinite groups, including free groups, free abelian groups, Artin groups, pure braid groups, Baumslag-Solitar groups, and infinite virtually cyclic groups. Furthermore, the contractibility is inherited by finite index subgroups of these groups, and preserved by their free products, direct products, and semidirect products.
The foundational results on subgroup graphs are straightforward and for the most part follow Kapovich and Myasnikov [loc. cit.]. It’s surprising that this generalization has never been developed as it is here.
For each such group \(G\) with presentation \(\langle\, X \mid R\,\rangle\) and each subgroup \(H\) of \(G\) of finite index \(n\), she associates a connected graph with \(n\) vertices whose edges are labeled by elements of \(X\) and whose closed paths reflect the relators \(R\). This subgroup graph, which is revealed to be the Schreier coset graph of \(H\) with respect to \(X\), carries much essential information about \(H\), and is unique up to isomorphism in this category. Furthermore, she identifies \(H\) as the language subgroup (in the sense of graph automata) of the subgroup graph. A first collection of applications describe how various properties of subgroups – for example, the conjugacy of subgroups and the normality of a subgroup – are reflected in their subgroup graphs.
Note that the main results on subgroup graphs apply to finite groups and their subgroups. The author shows how subgroup graphs identify Sylow (and, more generally, Hall) subgroups of a finite group, and uses these tools to prove an easy (and known) divisibility property involving a finite group with a malnormal subgroup (i.e. a group with a Frobenius kernel).
The second application relates to the poset \(P_{\mathrm{fi}}(G)\) of cosets of proper finite index subgroups of \(G\) under inclusion and its associated simplicial complex of chains \(\Delta(P_{\mathrm{fi}}(G))\). When \(G\) is finite, \(\Delta(P_{\mathrm{fi}}(G))\) is known not to be contractible. The author exhibits sufficient conditions for contractibility of \(\Delta(P_{\mathrm{fi}}(G))\) for a finitely generated group in terms of the existence of a family of subgroup graphs of selected finite index subgroups of \(G\). She then verifies this condition for a number of families of infinite groups, including free groups, free abelian groups, Artin groups, pure braid groups, Baumslag-Solitar groups, and infinite virtually cyclic groups. Furthermore, the contractibility is inherited by finite index subgroups of these groups, and preserved by their free products, direct products, and semidirect products.
The foundational results on subgroup graphs are straightforward and for the most part follow Kapovich and Myasnikov [loc. cit.]. It’s surprising that this generalization has never been developed as it is here.
Reviewer: Alberto Luis Delgado (Normal)
MSC:
| 20F05 | Generators, relations, and presentations of groups |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 05E45 | Combinatorial aspects of simplicial complexes |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20E07 | Subgroup theorems; subgroup growth |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
combinatorial group theory; finitely generated groups; finite index subgroups; presentations; subgroup graph; order and nerve complex; coset posetCitations:
Zbl 1001.20015References:
| [1] | Abels, Herbert; Holz, Stephan, Higher generation by subgroups, J. Algebra, 160, 2, 310-341 (1993), MR1244917 · Zbl 0811.20036 |
| [2] | Björner, A., Topological methods, (Handbook of Combinatorics, vols. 1, 2 (1995)), 1819-1872, MR1373690 · Zbl 0851.52016 |
| [3] | Brown, Kenneth S., The coset poset and probabilistic zeta function of a finite group, J. Algebra, 225, 2, 989-1012 (2000), MR1741574 · Zbl 0973.20016 |
| [4] | Kapovich, Ilya; Myasnikov, Alexei, Stallings foldings and subgroups of free groups, J. Algebra, 248, 2, 608-668 (2002), MR1882114 · Zbl 1001.20015 |
| [5] | Liebeck, Martin W.; Shalev, Aner, Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks, J. Algebra, 276, 2, 552-601 (2004), MR2058457 · Zbl 1068.20052 |
| [6] | Ramras, Daniel A., Connectivity of the coset poset and the subgroup poset of a group, J. Group Theory, 8, 6, 719-746 (2005), MR2179666 · Zbl 1096.20024 |
| [7] | Stallings, John R., Topology of finite graphs, Invent. Math., 71, 3, 551-565 (1983), MR695906 · Zbl 0521.20013 |
| [8] | Shareshian, John; Woodroofe, Russ, Order complexes of coset posets of finite groups are not contractible, Adv. Math., 291, 758-773 (2016), MR3459029 · Zbl 1348.20024 |
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.
