‎‎in cite{fr 1‎, ‎fr 2} using faithful complex representations of cyclically pinched and conjugacy pinched one-relator groups we proved that any limit group has a faithful representation in $psl(2,c)$‎. ‎further this representation can be effectively constructed using the jsj decomposition‎. ‎in this note we show that any hyperbolic cyclically pinched one-relator group with maximal amalgamated ...

recently‎, ‎the authors gave some conditions under which a direct product‎ ‎of finitely many groups is $mathcal{v}-$capable if and only if each of its‎ ‎factors is $mathcal{v}-$capable for some varieties $mathcal{v}$‎. ‎in this paper‎, ‎we extend this fact to any infinite direct product of groups‎. ‎moreover‎, ‎we conclude some results for $mathcal{v}-$capability of direct products of infinitel...

We show that an effective action of the one-dimensional torus $${\mathbb G}_m$$ on a normal affine algebraic variety X can be extended to semi-direct product G}_m\rightthreetimes {\mathbb G}_a$$ with same general orbit closures if and only there is divisor D consists -fixed points. This result applied study orbits automorphism group $${{\,\mathrm{Aut}\,}}(X)$$ X.