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Normalising graphs of groups

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Abstract

We discuss a partial normalisation of a finite graph of finite groups \((\Gamma (-), X)\) which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include: (1) an important inequality for the number of edges in a Stallings decomposition \(\Gamma \cong \pi _1(\Gamma (-), X)\) of a finitely generated virtually free group, (2) the proof of equivalence of a number of conditions for such a group to be ‘large’, as well as (3) the classification up to isomorphism of virtually free groups of (free) rank 2. We also discuss some number-theoretic consequences of the last result.

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Notes

  1. See Sect. 2 for the definition of the free rank.

  2. See [13, Cor. 1], or [4, Prop. 1] for a more general result.

  3. The notation used in Eq. (3.1) follows Serre; see Déf. 8 in [19, Sec. 4.4].

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    C. Krattenthaler

  2. School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London, E1 4NS, UK

    Thomas Müller

Authors
  1. C. Krattenthaler
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  2. Thomas Müller
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Correspondence to Thomas Müller.

Additional information

Communicated by J. S. Wilson.

C. Krattenthaler research partially supported by the Austrian Science Foundation FWF, grant S50-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. T. Müller research supported by Lise Meitner Grant M1661-N25 of the Austrian Science Foundation FWF.

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Krattenthaler, C., Müller, T. Normalising graphs of groups. Monatsh Math 185, 269–286 (2018). https://doi.org/10.1007/s00605-016-0992-z

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