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Separable and tree-like asymptotic cones of groups
Using methods from nonstandard analysis, we will discuss which metric spaces can be realized as ultralimits (in particular, asymptotic cones). For example, we will show that any separable ultralimit is proper. Applying the results we will find in the context of groups, we will classify the real trees appearing as asymptotic cones of (not necessarily hyperbolic) finitely generated groups. Also, we show that all proper metric spaces can be realized as asymptotic cones.