scholarly journals The A-T-menability of some graphs of groups with cyclic edge groups

2016 ◽  
Vol 163 (1) ◽  
pp. 145-159 ◽  
Author(s):  
MATHIEU CARETTE ◽  
DANIEL T. WISE ◽  
DANIEL J. WOODHOUSE
Keyword(s):  
Vertex Group ◽  
Graphs Of Groups

AbstractWe show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on ℍn.

scholarly journals Graphical splittings of Artin kernels

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrique Miguel Barquinero ◽  
Lorenzo Ruffoni ◽  
Kaidi Ye
Keyword(s):  
Free Group ◽  
Artin Group ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Block Graphs ◽  
Right Angled Artin Groups

Abstract We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

scholarly journals A criterion for residual 𝑝-finiteness of arbitrary graphs of finite 𝑝-groups

2019 ◽  
Vol 22 (5) ◽  
pp. 837-844
Author(s):  
Gareth Wilkes
Keyword(s):  
Fundamental Group ◽  
Infinite Graphs ◽  
Graphs Of Groups ◽  
Arbitrary Graphs

Abstract We establish conditions under which the fundamental group of a graph of finite p-groups is necessarily residually p-finite. The technique of proof is independent of previously established results of this type, and the result is also valid for infinite graphs of groups.

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

1992 ◽  
Vol 35 (3) ◽  
pp. 390-399 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Nilpotent Groups ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Isolated Subgroup ◽  
Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

scholarly journals Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sam Shepherd ◽  
Daniel J. Woodhouse
Keyword(s):  
Free Group ◽  
Large Family ◽  
Finitely Generated ◽  
Free Vertex ◽  
Cyclic Subgroups ◽  
Graphs Of Groups ◽  
Jsj Decomposition ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Abelian Subgroups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

scholarly journals On the one-endedness of graphs of groups

2015 ◽  
Vol 278 (2) ◽  
pp. 463-478 ◽  
Author(s):  
Nicholas Touikan
Keyword(s):  
Graphs Of Groups ◽  
The One

A Note on the Graphs of Groups, I

1972 ◽  
Vol 45 (1) ◽  
pp. 45-45
Author(s):  
Billy E. Milner
Keyword(s):  
Graphs Of Groups

scholarly journals Rigid cohomology of topological groupoids

1978 ◽  
Vol 26 (3) ◽  
pp. 277-301 ◽  
Author(s):  
K. A. MacKenzie
Keyword(s):  
Vector Bundles ◽  
Cohomology Theory ◽  
Unexpected Result ◽  
Vertex Group ◽  
Topological Groups ◽  
Locally Compact ◽  
Cohomology Space ◽  
Rigid Cohomology ◽  
Topological Groupoids

AbstractA cohomology theory for locally trivial, locally compact topological groupoids with coefficients in vector bundles is constructed, generalizing constructions of Hochschild and Mostow (1962) for topological groups and Higgins (1971) for discrete groupoids. It is calculated to be naturally isomorphic to the cohomology of the vertex groups, and is thus independent of the twistedness of the groupoid. The second cohomology space is accordingly realized as those “rigid” extensions which essentially arise from extensions of the vertex group; the cohomological machinery now yields the unexpected result that in fact all extensions, satisfying some natural weak conditions, are rigid.

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