A combination theorem for cubulation in small cancellation theory over free products

Martin, Alexandre; Steenbock, Markus

doi:10.5802/aif.3118

Cited by 13 publications

(19 citation statements)

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“…By the aforementioned results of Haglund [14], G cannot act properly on a CAT(0) cube complex. A recent result of Martin and Steenbock shows that every finitely presented classical C * ( 1 6 )-group acts properly cocompactly on a CAT(0) cube complex if every generating free factor does [17]. Our example shows that this does not extend in any way to infinite presentations.…”

Section: Proof Of Theorem 31mentioning

confidence: 66%

Infinitely presentedC(6)-groups are SQ-universal

Gruber

2015

J. London Math. Soc.

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Section: Proof Of Theorem 31mentioning

confidence: 66%

Infinitely presentedC(6)-groups are SQ-universal

Gruber

2015

J. London Math. Soc.

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“…These groups satisfy the so-called C I p 1 6 q small-cancellation condition when n ¥ 6, so this result is also covered in [Wis04]. An extension of Wise's result for C I p 1 6 q groups was pursued by Martin and Steenbock in 2014 when they successfully cubulated C I p 1 6 q small cancellation free products of cubulable groups [MS17] (see also [JW17]). In this thesis, we generalize Lauer and Wise's cubulation results for one-relator groups with torsion to the free product setting.…”

Section: Summary Of Resultsmentioning

confidence: 95%

Cubulating one-relator products with torsion

Stucky

2021

Groups Geom. Dyn.

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Since the resolution of the virtual Haken conjecture in the theory of hyperbolic 3-manifolds, there has been much attention devoted to CATp0q cube complexes. These non-positively curved metric spaces are powerful tools for understanding innite, nitely generated groups in part because of their cubical combinatorics. Simply knowing that a group is cubulable (acts geometrically properly and cocompactly by isometries on a CATp0q cube complex) is sucient to unlock a good deal of structural information about it, and cubulating groups has become an important goal of modern geometric group theory.In 2013, Lauer and Wise showed that a one-relator group with torsion whose dening relator has exponent at least 4 is cubulable. To achieve this, they build a system of nicely-behaved codimension-1 subspaces (walls) in the universal cover and invoke a construction due to Sageev.In this thesis, we achieve a generalization of this result to one-relator products with torsion, namely, that a one-relator product of locally indicable groups whose dening relator has exponent at least 4 admits a geometric action on a CATp0q cube complex if the factors do. Our results are framed in the more general context of staggered quotients of free products of nitely many locally indicable and cubulable groups. The main tools are geometric small-cancellation results for van Kampen diagrams over these groups, which allow us to argue that walls are plentiful and geometrically well-behaved in the universal cover. Relative hyperbolicity of these one-relator products and relative quasiconvexity of wall stabilizers both play a central role. Using Agol's theorem that a hyperbolic, cubulable group is virtually special, we obtain as a corollary that the one-relator products we consider are virtually special provided that the factors are hyperbolic in addition to the other assumptions.xi

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“…Martin and Steenbock recently showed that a small-cancellation quotient of a free product of cubulated groups is cubulated [MS17]. In this paper we revisit their theorem in a slightly weaker form, and reprove it in a manner that capitalizes on the available technology.…”

Section: Introductionmentioning

confidence: 95%