scholarly journals Neighbourhood Graphs of Cayley Graphs for Finitely-generated Groups

2002 ◽  
Vol 23 (6) ◽  
pp. 733-740
Author(s):  
Markus Neuhauser
Keyword(s):  
Cayley Graphs ◽  
Finitely Generated ◽  
Finitely Generated Groups

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

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 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.

The word problem for semigroups satisfying x3 = x

1978 ◽  
Vol 84 (1) ◽  
pp. 11-19 ◽  
Author(s):  
J. A. Gerhard
Keyword(s):  
Word Problem ◽  
Upper Bound ◽  
Finitely Generated ◽  
Finitely Generated Groups

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

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