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A presentation for a group of automorphisms of a simplicial complex

  • M. A. Armstrong

Extract

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

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References

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1.Armstrong, M. A., Trees, tail wagging and group presentations, L'Enseignement Mathematique 32 (1986), 261270.
2.Brown, K. S., Presentations for groups acting on simply connected complexes, J. Pure and Applied Algebra 32 (1984), 110.
3.Serre, J.-P., Arbres, Amalgames, SL2, Astérisque 46 (Soc. Math, de France 1977).

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