Non zariski dense embeddings of a free group into a Lie group

From: Mark Zorro (markzorromz_at_yahoo.com)
Date: 07/22/04

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    Date: 22 Jul 2004 14:13:16 -0700

    Consider the Lie group SL(2, C), where C is the complex field. It has
    come to my attention that most random pairs (r_1, r_2) of matrices
    taken from SL(2,C) will generate a Zariski dense free group of rank
    two in SL(2,C). My question is the following:

    1) Are there examples of embeddings \rho of the rank two free group,
    F_2, into SL(2,C) having the property that the Zariski closure of
    \rho(F_2) is one dimensional in SL(2,C)? I dont even have an example
    of an embedding of F_2 into SL(2,C) with the property that the Zariski
    closure of the corresponding points is two dimensional.

    M. Zorro

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