Presentation complex

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In geometric group theory, a presentation complex is a 2-dimensional cell complex associated to any presentation of a group G. The complex has a single vertex, and one loop at the vertex for each generator of G. There is one 2-cell for each relation in the presentation, with the boundary of the 2-cell attached along the appropriate word.

[edit] Properties

The fundamental group of the presentation complex is the group G itself.
The universal cover of the presentation complex is a Cayley complex for G, whose 1-skeleton is the Cayley graph of G.
Any presentation complex for G is the 2-skeleton of an Eilenberg-Maclane space K(G,1).

[edit] Example

Let G =Z² be the two-dimensional integer lattice, with a presentation

$G=\langle x,y|xyx^{-1}y^{-1}\rangle.$

Then the presentation complex for G is a torus, obtained by gluing the opposite sides of a square, the 2-cell, which are labelled x and y. All four corners of the square are glued into a single vertex, the 0-cell of the presentation complex, while a pair consisting of a longtitudal and meridian circles on the torus, intersecting at the vertex, constitutes its 1-skeleton.

The associated Cayley complex is a regular tiling of the plane by unit squares. The 1-skeleton of this complex is a Cayley graph for Z².

[edit] References

Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory. Reprint of the 1977 edition (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89). Classics in Mathematics. Springer-Verlag, Berlin, 2001 ISBN 3-540-41158-5