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**11 - 11**of**11**### ACTION TYPE GEOMETRICAL EQUIVALENCE OF REPRESENTATIONS OF GROUPS.

, 2008

"... In the paper we prove (Theorem 8.1) that there exists a continuum of non isomorphic simple modules over KF2, where F2 is a free group with 2 generators (compare with [Ca] where a continuum of non isomorphic simple 2-generated groups is constructed). Using this fact we give an example of a non action ..."

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In the paper we prove (Theorem 8.1) that there exists a continuum of non isomorphic simple modules over KF2, where F2 is a free group with 2 generators (compare with [Ca] where a continuum of non isomorphic simple 2-generated groups is constructed). Using this fact we give an example of a non action type logically Noetherian representation (Section 9). In general, the topic of this paper is the action type algebraic geometry of representations of groups. For every variety of algebras Θ 1 and every algebra H ∈ Θ we can consider an algebraic geometry in Θ over H. Algebras in Θ may be many sorted (not necessarily one sorted) algebras. A set of sorts Γ is fixed for each Θ. This theory can be applied to the variety of representations of groups over fixed commutative ring K with unit. We consider a representation as two sorted algebra (V,G), where V is a K-module, and G is a group acting on