Funding: Acknowledgements. The authors would also like to thank the referee for their very careful and thorough reading of paper. This publication is in part a product visit first third author Mathematisches Forschungsinstitut Oberwolfach, Germany December 2016 as Research In Pairs program. was partially supported by NSF grants DMS–1600635 DMS-1854367.

A group is called morphic if for each normal endomorphism α in end(G),there exists β such that ker(α)= Gβ and Gα= ker(β). In this paper, we consider the case that there exist normal endomorphisms β and γ such that ker(α)= Gβ and Gα = ker(γ). We call G quasi-morphic, if this happens for any normal endomorphism α in end(G). We get the following results: G is quasi-morphic if and only if, for any ...

The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element, g2 HzL = -1 ê z, and an element of infinite order, g• HzL = z + 1. Viewing the action of the group elements on a...