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Presentation complex

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Source updated August 5, 2020

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.

01Properties

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 $K(G,1)$ .

02Examples

Let $G=\mathbb {Z} ^{2}$ be the two-dimensional integer lattice, with 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 $\mathbb {Z} ^{2}$ .

Let $G=\mathbb {Z} _{2}*\mathbb {Z} _{2}$ be the Infinite dihedral group, with presentation $\langle a,b\mid a^{2},b^{2}\rangle$ . The presentation complex for $G$ is $\mathbb {RP} ^{2}\vee \mathbb {RP} ^{2}$ , 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.

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Sources and credits

This article is adapted from the Wikipedia article “Presentation complex”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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