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

TL;DR

This paper explores functions satisfying the parallelogram identity on finitely generated groups, linking them to tangent vectors at the trivial character in SL_2(C) character varieties, and studies deformation obstructions based on the group's first homology dimension.

Contribution

It characterizes the space of parallelogram identity functions on groups and analyzes the smoothness of the trivial character in the character variety based on homology.

Findings

Functions satisfying the parallelogram identity correspond to tangent vectors at the trivial character.

The trivial character is smooth if dim H_1(G,C)<2.

The trivial character is not smooth if dim H_1(G,C)>2.

Abstract

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 these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstructions for deforming the trivial character in the direction given by f. Along the way, we show that the trivial character is a smooth point of the character variety if dim H_1(G,C)<2 and not a smooth point if dim H_1(G,C)>2.