Bredon cohomological finiteness conditions for generalisations of\n Thompson's groups

2011/05/01 by Conchita Martı́nez-Pérez, Conchita Martinez-Perez, Martinez-Perez, Conchita +2
Mathematics · #20J05 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:20J05

paper · pdf · doi:10.48550/arxiv.1105.0189

22 pages, to appear Groups, Geometry, and Dynamics

openalex publication_date 2011/05/01 · arxiv created 2013/09/09 · arxiv updated 2013/09/10 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28

Abstract

We define a family of groups that generalises Thompson's groups T and G\nand also those of Higman, Stein and Brin. For groups in this family we descrine\ncentralisers of finite subgroups and show, that for a given finite subgroup\nQ, there are finitely many conjugacy classes of finite subgroups isomorphic\nto Q. We use this to show that whenever defined, the T versions of these\ngroups have a slightly weaker property, quasi- underline operatorname\nF_\∞, to that of a group possessing a finite type model for the\nclassifying space for proper actions underlineEG if and only if they\nposses finite type models for the ordinary classifying space. We also\ngeneralise some well-known properties of ordinary cohomology to Bredon\ncohomology.\n

Citations

• Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms

Related

• Cohomological finiteness conditions and centralisers in generalisations of Thompson's group V
• Fixed points of finite groups acting on generalised Thompson groups
• Cohomological finiteness properties of the Brin-Thompson-Higman groups 2V and 3V
• Euler classes and Bredon cohomology for groups with restricted families of finite subgroups
• Towards computing the rational homology and assembly maps of generalised Thompson groups