Hyperbolic generalized triangle groups, property (T) and finite simple quotients

We construct several series of explicit presentations infinite hyperbolic groups enjoying Kazhdan's property (T). Some them are significantly shorter than the previously known shortest examples. Moreover, we show that some those Kazhdan possess finite simple quotient arbitrarily large rank; they constitute first-known specimens combining properties. All consider non-positively curved k $k$ -fold generalized triangle groups, is, a simplicial action on CAT(0) complex, which is sharply transitive set triangles, and such edge-stabilizers cyclic order . Appendices A, B C provided separately as supplementary material with published article.