57 research outputs found
Finite graphs and amenability
Hyperfiniteness or amenability of measurable equivalence relations and group
actions has been studied for almost fifty years. Recently, unexpected
applications of hyperfiniteness were found in computer science in the context
of testability of graph properties. In this paper we propose a unified approach
to hyperfiniteness. We establish some new results and give new proofs of
theorems of Schramm, Lov\'asz, Newman-Sohler and Ornstein-Weiss
Convergence and limits of linear representations of finite groups
Motivated by the theory of graph limits, we introduce and study the
convergence and limits of linear representations of finite groups over finite
fields. The limit objects are infinite dimensional representations of free
groups in continuous algebras. We show that under a certain integrality
condition, the algebras above are skew fields. Our main result is the extension
of Schramm's characterization of hyperfiniteness to linear representations.Comment: Final version. To appear in the Journal of Algebr
The Strong Approximation Conjecture holds for amenable groups
We prove that the approximation conjecture of Luck holds for all amenable
groups in the complex group algebra case. This result was previously proved by
Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the
group is torsion-free elementary amenable
The amenability of affine algebras
We introduce the notion of amenability for affine algebras. We characterize
amenability by Folner-sequences, paradoxicality and the existence of finitely
invariant dimension-measures. Then we extend the results of Rowen on ranks,
from affine algebras of subexponential growth to amenable affine algebras
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