In this paper, we briefly introduce different types of buildings such as spherical buildings, Euclidean buildings, twin buildings, R-buildings and describe some of their applications to many subjects: (1) differential geometry such as Mostow strong rigidity, rank rigidity of nonpositively curved manifolds, Margulis superrigidity, quasi-isometry rigidity, classification of isoparametric manifold...

In the past few decades there has been considerable interest in word maps on groups with emphasis (non-abelian) finite simple groups. Various asymptotic results (holding for sufficiently large groups) have obtained. More recently non-asymptotic all emerged, particular words (commutators and certain power words) which are not an identity of any group. this paper we initiate a systematic study ab...