Consider a random group \(\Gamma\) with \(k\) generators and \(r\) random relators of large length \(N\). We ask about the geometry of the character variety of \(\Gamma\) with values in \(\mathrm{SL}(2,\mathbb{C})\) or any semisimple Lie group \(G\). This is the moduli space of group homomorphisms from \(\Gamma\) to \(G\) up to conjugation. We show that with an exponentially small proportion of exceptions the character variety is empty, \(k\lt r+1\), finite and large, \(k=r+1\), or irreducible of dimension \((k-r-1) \mathrm{dim}\thinspace G\), \(k\gt r+1\). The proofs use new results on expander graphs for finite simple groups of Lie type and are conditional of the Riemann hypothesis.
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Passcode: A group is called an ________ group if it admits an invariant mean. (8 letters, lowercase)