The positive first order theory of a group G is the set of all sentences only involving equalities (no negation is allowed) that are satisfied in G. All non-abelian free groups have the same positive first order theory, moreover this theory is contained in the the positive first-order theory of every other group G. When equality holds, we say that G has trivial positive first-order theory. Of course not all groups have trivial positive first-order theory. However until now all known examples of such groups contains non-abelian free subgroups. In this talk we will explain how geometric / topological tools coming from small cancellation theory and action on R-trees can be used to produce some surprising groups. Although they have trivial positive first-order theory they satisfy some other "pathological" properties : they cannot act in a non-degenerated way on a hyperbolic space; they are simple, torsion-free groups, all of whose proper subgroups are cyclic; they have no unbounded quasi-morphisms; etc. This is joint work with Francesco Fournier-Facio and Turbo Ho.