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Wikipedia / 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.

Properties

Examples

be the two-dimensional integer lattice, with presentation

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.

be the Infinite dihedral group, 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.

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

  • Ronald Brown and Johannes 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, Cynthia, Metzler, Wolfgang and Sieradski, Allan J. (eds.). Two-dimensional homotopy and combinatorial group theory, London Mathematical Society Lecture Note Series, Volume 197. Cambridge University Press, Cambridge (1993).