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A basic consequence of the Seifert–van Kampen theorem is the following. Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...

A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...