The parallelogram identity on groups and deformations of the trivial character in SL_2(C)

Julien Marché, Maxime Wolff

We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that t…