LAUSR

Let $$\varGamma $$Γ be a finitely generated group and G a real form of $$\mathrm {SL}_n(\mathbb {C})$$SLn(C). We propose a definition for the G-character variety of $$\varGamma $$Γ as a subset of the $$\mathrm {SL}_n(\mathbb {C})$$SLn(C)-character variety of $$\varGamma $$Γ. We consider two anti-holomorphic involutions of the $$\mathrm {SL}_n(\mathbb {C})$$SLn(C) character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of $$\mathrm {SL}_n(\mathbb {C})$$SLn(C). We study in detail an example: the $$\mathrm {SL}_n(\mathbb {C})$$SLn(C), $$\mathrm {SU}(2,1)$$SU(2,1) and $$\mathrm {SU}(3)$$SU(3) character varieties of the free product $$\mathbb {Z}/{3}\mathbb {Z}*\mathbb {Z}/{3}\mathbb {Z}$$Z/3Z∗Z/3Z.