Assess students' ability to design and construct a new infinite family of expander graphs realized as Cayley graphs of finite quotients of a fixed residually finite group; require an original explicit construction, rigorous proofs of expansion via quantitative spectral gap bounds (e.g., lower bounds on the first nontrivial eigenvalue or Cheeger constant), and a detailed analysis of diameter growth and the eigenvalue distribution asymptotics. Expect use of advanced group theory (residual finiteness, profinite quotients), representation-theoretic or geometric methods (property (T)/(τ) where relevant), spectral graph theory, and analytic estimates; solutions must state assumptions, provide explicit constants or effective bounds, and include proof sketches or full arguments as appropriate.