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The fundamental theorem of cubical small cancellation theory. (English) Zbl 1365.20034
Small 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.

MSC:
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
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[6] mixed Piotr Przytycki and Daniel T. Wise, \newblock Mixed manifolds are virtually special, \newblock \urlhttp://arxiv.org/abs/1205.6742.
[7] \bibtartakovskiiarticle author=Tartakovski\u \i , V. A., title=Solution of the word problem for groups with a \(k\)-reduced basis for \(k>6\), journal=Izvestiya Akad. Nauk SSSR. Ser. Mat., volume=13, pages=483–494, date=1949, language=Russian, issn=0373-2436, review=\MR 0033816,
[8] hierarchy Daniel T. Wise, \newblock The structure of groups with quasiconvex hierarchy, \newblock \urlhttps://docs.google.com/open?id=0B45cNx80t5-2T0twUDFxVXRnQnc (see ).\pagebreak
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[10] Wise, Daniel T., From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics 117, xiv+141 pp., (2012), published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI · Zbl 1278.20055
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