UC Irvine

We adapt a partial notion of EF games to a continuous logic for unbounded pointed metric structures, and use this to investigate the elementary equivalence of certain metric structures. Particular focus is placed upon EF games between asymptotic cones of symmetric spaces $X$ arising from semisimple Lie groups. Kramer, et al. showed that, depending on the truth of CH, there is, up to homeomorphism, either $1$ or $2^{2^{\aleph_0}}$-many asymptotic cones of $X$ as one varies the choice of ultrafilter. This leaves open the possibility that all such asymptotic cones are elementarily equivalent. Towards a proof of the elementary equivalence, we utilize the fact that these asymptotic cones are known to be isometric to the point spaces of certain nondiscrete affine $\R$-buildings. We investigate the building structure and demonstrate the elementary equivalence of parallel classes of walls, which are fundamental to the classification of affine $\R$-buildings.