scholarly journals A constructive way to compute the Tarski number of a group

2018 ◽  
Vol 17 (07) ◽  
pp. 1850139
Author(s):  
Akram Yousofzadeh
Keyword(s):  
Upper Bound ◽  
Amenable Group ◽  
Combinatorial Property ◽  
Paradoxical Decompositions

The Tarski number of a group [Formula: see text] is the minimal number of the pieces of paradoxical decompositions of that group. Using configurations along with a matrix combinatorial property, we construct paradoxical decompositions. We also compute an upper bound for the Tarski number of a given non-amenable group by counting the number of paths in a diagram associated to the group.

scholarly journals An upper bound for the Tarski numbers of nonamenable groups of piecewise projective homeomorphisms

2017 ◽  
Vol 27 (03) ◽  
pp. 315-321 ◽  
Author(s):  
Yash Lodha
Keyword(s):  
Upper Bound ◽  
Amenable Group ◽  
Affine Map ◽  
Finitely Presented Groups ◽  
Free Subgroups ◽  
Paradoxical Decomposition ◽  
Finitely Presented

The Tarski number of a nonamenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Nonamenable groups of piecewise projective homeomorphisms were introduced in [N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. 110(12) (2013) 4524–4527], and nonamenable finitely presented groups of piecewise projective homeomorphisms were introduced in [Y. Lodha and J. T. Moore, A finitely presented non amenable group of piecewise projective homeomorphisms, Groups, Geom. Dyn. 10(1) (2016) 177–200]. These groups do not contain non-abelian free subgroups. In this paper, we prove that the Tarski number of all groups in both families is at most 25. In particular, we demonstrate the existence of a paradoxical decomposition with 25 pieces. Our argument also applies to any group of piecewise projective homeomorphisms that contains as a subgroup the group of piecewise [Formula: see text] homeomorphisms of [Formula: see text] with rational breakpoints and an affine map that is a not an integer translation.

scholarly journals Extensions of Positive Definite Functions on Amenable Groups

2011 ◽  
Vol 54 (1) ◽  
pp. 3-11 ◽  
Author(s):  
M. Bakonyi ◽  
D. Timotin
Keyword(s):  
Cayley Graph ◽  
Amenable Group ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Combinatorial Property ◽  
Definite Function ◽  
Amenable Groups ◽  
Definite Operator ◽  
New Applications

AbstractLet S be a subset of an amenable group G such that e ∈ S and S–1 = S. The main result of this paper states that if the Cayley graph of G with respect to S has a certain combinatorial property, then every positive definite operator-valued function on S can be extended to a positive definite function on G. Several known extension results are obtained as corollaries. New applications are also presented.

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