# Presentation complex

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.

## Examples

Let ${\displaystyle G=\mathbf {Z} ^{2}}$ be the two-dimensional integer lattice, with a presentation

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

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 ${\displaystyle \mathbf {Z} ^{2}}$

Let ${\displaystyle G=\mathbf {Z} _{2}*\mathbf {Z} _{2}}$ be the Infinite dihedral group, with presentation ${\displaystyle \langle a,b\mid a^{2},b^{2}\rangle }$. The presentation complex for ${\displaystyle G}$ is ${\displaystyle \mathbb {R} P^{2}\vee \mathbb {R} P^{2}}$, the wedge sum of projective planes. For each path, there is one 2-cell glued to each loop, which provides the standard cell structure for each projective plane.

The Cayley Complex is an infinite string of spheres.[1]

## References

1. ^ Hatcher, Allen (2001-12-03). Algebraic Topology (1st ed.). Cambridge: Cambridge University Press. ISBN 9780521795401.
• 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
• R. Brown and J. Huebschmann, Identities among relations, in Low dimensional topology, London Math. Soc. Lecture Note Series 48 (ed. R. Brown and T.L. Thickstun, Cambridge University Press, 1982), pp. 153–202.
• Hog-Angeloni, C., Metzler, W. and Sieradski, A.~J. (eds.). Two-dimensional homotopy and combinatorial group theory, London Mathematical Society Lecture Note Series, Volume 197. Cambridge University Press, Cambridge (1993).