This is a detailed expository paper of the article “Random walk on random groups” by M. Gromov [ibid. 13, No. 1, 73-146 (2003; Zbl 1122.20021)]. Let \(\Gamma\) be a finitely generated group with symmetric generator set \(S\) of size \(2k\), \(G=(V,E)\) be a graph and \(\overrightarrow G\) be the directed graph associated to \(G\). Let \(\alpha\colon\overrightarrow E\ni\overrightarrow e\mapsto\alpha(\overrightarrow e)\in S\) be a given map, and extend it to a path \(\overrightarrow p\) in \(\overrightarrow G\). We assume that \(\alpha\) is symmetric. Let \(R_\alpha=\{\alpha(\overrightarrow p):\overrightarrow p\) is a cycle in \(\overrightarrow G\}\), and \(W_\alpha\) be the normal closure of \(R_\alpha\) in \(\Gamma\). Put \(\Gamma_\alpha=\Gamma/W_\alpha\). Choose \(\alpha(\overrightarrow e)\) uniformly and independently from \(S\). Then \(\Gamma_\alpha\) is our random group. Put \(X_\alpha=\text{Cayley}(\Gamma_\alpha,S)\). \(G\) acts on \(\Gamma_\alpha\) and \(X_\alpha\) by \(\alpha(\overrightarrow e)\).
The main result is Theorem: If \(G\) is an expander, \(3\leq\deg(u)\leq d\) for all \(u\in V\), and the girth of \(G\) is large enough, then \(\Gamma_\alpha\) has property (T) with high probability. So, every unitary representation of the group \(\Gamma_\alpha\) has a fixed vector.