2021/11/04 by Eduard Einstein, Einstein, Eduard, Thomas Ng +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2111.03008
openalex publication_date 2021/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under C'(\frac16)--small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on CAT(0) cube complexes. Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion and proving that wall stabilizers form a sufficiently rich family of full relatively quasiconvex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. In particular, we prove that C'(\frac16)--small cancellation free products of residually finite groups are residually finite.