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Test sequences and formal solutions over hyperbolic groups

Given a group G and a sentence $\sigma$ in the language of groups, we say that $\sigma$ is a true sentence in G (denoted by $G\models \sigma$) if $\sigma$ is true if interpreted in G. The first-order or elementary theory of a group G, denoted by Th(G), is the set of all sentences true in G. The groups...