In the first half of this talk, I will construct a special family of trivalent expander graphs from a infinite group acting on a particular simplicial complex which is a Euclidean building. Expander graph families are not only theoretically interesting objects with many relations to other areas in pure mathematics (like, e.g., spectral geometry, representation theory, number theory or random walks) but have also practical importance (e.g., for robust networks, complexity theory or the constuction of efficient codes). In the second half of this talk, I will discuss how this family of expander graphs gives rise to associated surfaces of constant curvature minus one with uniform lower bounds on their first nontrivial Laplace eigenvalues, using results of Buser and Brooks.

This talk is based on the joint work with Ioannis Ivrissimtzis and Alina Vdovina.

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