The fundamental theorem of cubical small cancellation theory.

*(English)*Zbl 1365.20034Small cancellation theory was initiated by V. A. Tartakovskij [Izv. Akad. Nauk SSSR, Ser. Mat. 13, 483–494 (1949; Zbl 0035.29501)], following earlier work of Dehn and others. Cubical small cancellation theory is a generalization of the classical results, due to D. T. Wise [Electron. Res. Announc. Math. Sci. 16, 44–55 (2009; Zbl 1183.20043)] and builds upon the theory of non-positively curved cube complexes. The aims of the current paper are varied (we refer the reader to the paper for appropriate definitions and terminology). After a brief introduction and exposition of classical small cancellation theory, the author gives a new proof of the fundamental theorem that if \(X\) is a \(C(6)\)-complex and \(D\longrightarrow X\) a minimal disc diagram, then either \(D\) is a single cell, or \(D\) is a ladder or \(D\) has at least \(3\) spurs or shells of degree at most \(3\). The author then discusses non-positively curved cubic complexes and proves that if \(X\) is a non-positively curved cubic complex and \(D\longrightarrow X\) is a minimal disc diagram, then \(D\) is a path graph or it has at least \(3\) corners and/or spurs. He then provides an exposition fo cubical small cancellation theory and finally gives the proof of the main result, stating that if \((X, \{Y_i\})\) satisfies \(C(9)\) and \((D,\partial D)\longrightarrow (X^*,X)\) is a minimal disc diagram, then either \(D\) is a single vertex or single cone cell, or \(D\) is a ladder, or \(D\) has at least \(3\) shells of degree at most \(4\) and/or corners and/or spurs.

Reviewer: Martyn Dixon (Tuscaloosa)

##### MSC:

20F06 | Cancellation theory of groups; application of van Kampen diagrams |

20F65 | Geometric group theory |

20F67 | Hyperbolic groups and nonpositively curved groups |

##### Keywords:

small cancellation theory; cubical small cancellation theory; minimal disc diagram; ladder; non-positively curved##### References:

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