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 Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...

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 A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...

Properties

* The

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

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)

Examples

Let

G= \Z^2

be the two-dimensional integer lattice, with presentation :

G=\langle x,y, xyx^y^\rangle.

Then the presentation complex for ''G'' is a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

, 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 Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...

by unit squares. The 1-skeleton of this complex is a Cayley graph for

\Z^2

. Let

G = \Z_2 *\Z_2

be the

Infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', s ...

, with presentation

\langle a,b \mid a^2,b^2 \rangle

. The presentation complex for

G

\mathbb^2 \vee \mathbb^2

, the

wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...

of projective planes. For each path, there is one 2-cell glued to each loop, which provides the standard

cell structure Cell structure may refer to: * Cell (biology)#Anatomy * An organelle, or the layout of organelles of the biological cell itself * The structure of a covert cell, often involved in underground resistance, organised crime, terrorism or any group requ ...

for each projective plane. The Cayley complex is an infinite string of spheres.

References

Roger C. Lyndon Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.. He is known for Lyndon words, the Curtis–Hedlund–Lyndon theorem, Craig–Lyndon interpolation a ...

and

Paul E. Schupp Paul Eugene Schupp (born March 12, 1937, died January 24, 2022) was a professor emeritus of mathematics at the University of Illinois at Urbana Champaign. He is known for his contributions to geometric group theory, computational complexity and th ...

, ''Combinatorial group theory''. Reprint of the 1977 edition ( Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89). Classics in Mathematics.

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, 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). Algebraic topology Geometric group theory {{topology-stub