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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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References:
 [1] Greendlinger, Martin, On Dehn’s algorithms for the conjugacy and word problems, with applications, Comm. Pure Appl. Math., 13, 641-677, (1960) · Zbl 0156.01303 [2] Haglund, Fr\'ed\'eric; Wise, Daniel T., Special cube complexes, Geom. Funct. Anal., 17, 5, 1551-1620, (2008) · Zbl 1155.53025 [3] Kampen, Egbert R. Van, On some lemmas in the theory of groups, Amer. J. Math., 55, 1-4, 268-273, (1933) · Zbl 0006.39204 [4] Lyndon, Roger C.; Schupp, Paul E., Combinatorial group theory, Classics in Mathematics, xiv+339 pp., (2001), Springer-Verlag, Berlin · Zbl 0997.20037 [5] McCammond, Jonathan P.; Wise, Daniel T., Fans and ladders in small cancellation theory, Proc. London Math. Soc. (3), 84, 3, 599-644, (2002) · Zbl 1022.20012 [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 [9] Wise, D. T., Cubulating small cancellation groups, Geom. Funct. Anal., 14, 1, 150-214, (2004) · Zbl 1071.20038 [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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