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We describe on any finitely generated group G the space of maps G->C whichsatisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y).  It is known (but not well-known) that these functions correspond toZariski-tangent vectors at the trivial character of the character variety of Gin SL_2(C). We study the obstructions for deforming the trivial character inthe direction given by f. Along the way, we show that the trivial character isa smooth point of the character variety if dim H_1(G,C)<2 and not a smoothpoint if dim H_1(G,C)>2.

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