scholarly journals Linear and projective boundaries in HNN-extensions and distortion phenomena

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Bernhard Krön ◽  
Jörg Lehnert ◽  
Maya Stein
Keyword(s):  
Cayley Graphs ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Hnn Extensions

AbstractLinear and projective boundaries of Cayley graphs were introduced in [Glasg. Math. J. (2014), DOI 10.1017/S0017089514000512] as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbitsWe show that for all finitely generated groups, the distance between the antipodal pointsWe also exhibit a group with elements

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.

ON GROUPS WHICH CONTAIN NO HNN-EXTENSIONS

2007 ◽  
Vol 17 (07) ◽  
pp. 1377-1387 ◽  
Author(s):  
DEREK J. S. ROBINSON ◽  
ALESSIO RUSSO ◽  
GIOVANNI VINCENZI
Keyword(s):  
Large Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Rank 2 ◽  
Locally Graded Groups

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin.

A QUASI-ISOMETRY INVARIANT LOOP SHORTENING PROPERTY FOR GROUPS

2008 ◽  
Vol 18 (08) ◽  
pp. 1243-1257 ◽  
Author(s):  
STEPHEN G. BRICK ◽  
JON M. CORSON ◽  
DOHYOUNG RYANG
Keyword(s):  
Isoperimetric Inequality ◽  
Metric Spaces ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Quadratic Isoperimetric Inequality ◽  
Finitely Presented

We first introduce a loop shortening property for metric spaces, generalizing the property considered by M. Elder on Cayley graphs of finitely generated groups. Then using this metric property, we define a very broad loop shortening property for finitely generated groups. Our definition includes Elder's groups, and unlike his definition, our property is obviously a quasi-isometry invariant of the group. Furthermore, all finitely generated groups satisfying this general loop shortening property are also finitely presented and satisfy a quadratic isoperimetric inequality. Every CAT(0) cubical group is shown to have this general loop shortening property.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

scholarly journals The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

1992 ◽  
Vol 53 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Raptis ◽  
D. Varsos
Keyword(s):  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Hnn Extensions ◽  
Base Group ◽  
Generalized Free Products

AbstractWe study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

scholarly journals Some properties of the growth and of the algebraic entropy of group endomorphisms

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Pablo Spiga
Keyword(s):  
Finite Groups ◽  
Addition Theorem ◽  
Finitely Generated ◽  
Growth Type ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Algebraic Entropy ◽  
Finitely Generated Groups ◽  
Identity Automorphism ◽  
Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

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