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  "body": "\n## Definition\n\nA *graph* in the sense of Serre consists of a set $V$ of vertices and a set $E$ of oriented edges.\nIt comes with a map $e \\mapsto \\bar e$ from $E$ to $E$ with $\\bar{\\bar e} = e$ and $\\bar e \\neq e$, and maps $o, t \\colon E \\to V$ with $o(\\bar e) = t(e)$.\nA *graph of groups* $\\mathbb{G}$ consists of:\n\n1. a connected [[graph|00C0BR7D7H]] $Y$ in this sense;\n2. a [[group|G5S1M9]] $G_v$ for each vertex $v$;\n3. a group $G_e$ for each edge $e$, with $G_{\\bar e} = G_e$;\n4. an injective [[homomorphism|0JYTG5K9Y4]] $\\alpha_e \\colon G_e \\to G_{t(e)}$ for each edge $e$.\n\nLet $T$ be a maximal [[tree|062D9KXJB1]] in $Y$.\nForm the [[free product|TM7QS7P7X9]] of the groups $G_v$ and of the free group on symbols $t_e$, $e \\in E$.\nThe *fundamental group* $\\pi_1(\\mathbb{G}, T)$ is its quotient by the relations\n\n- $t_{\\bar e} = t_e^{-1}$ for each edge $e$;\n- $t_e \\, \\alpha_e(g) \\, t_e^{-1} = \\alpha_{\\bar e}(g)$ for each edge $e$ and each $g \\in G_e$;\n- $t_e = 1$ for each edge $e$ of $T$.\n\n## Remarks\n\nUp to isomorphism, $\\pi_1(\\mathbb{G}, T)$ does not depend on the choice of $T$, and each $G_v$ embeds in it.\n\nExample: $Y$ has two vertices $u, v$ and one pair of edges $e, \\bar e$ from $u$ to $v$, with $G_u = A$, $G_v = B$ and $G_e = C$.\nThen $T = Y$, all $t_e$ are trivial, and the relations identify $\\alpha_{\\bar e}(c) \\in A$ with $\\alpha_e(c) \\in B$; so $\\pi_1(\\mathbb{G}, T)$ is the amalgamated free product $A *_C B$.\n\nExample: $Y$ has one vertex $v$ and a loop $e$, with $G_v = A$ and $G_e = C$.\nThen $T = \\set{v}$, and $\\pi_1(\\mathbb{G}, T)$ is generated by $A$ and $t = t_e$ with $t \\, \\alpha_e(c) \\, t^{-1} = \\alpha_{\\bar e}(c)$; this is the [[HNN extension|YEYERRY37K]] of $A$ along the isomorphism $\\alpha_e(C) \\to \\alpha_{\\bar e}(C)$.\n\nNon-example: if some map $\\alpha_e$ is not injective, clause 4 fails, and the data is not a graph of groups; the vertex groups need not embed in the resulting quotient.\n\nWhen all vertex and edge groups are trivial, $\\pi_1(\\mathbb{G}, T)$ is the free group on the edges of $Y$ outside $T$, one generator for each pair $e, \\bar e$.\nThis is the fundamental group of $Y$ as a topological graph.\n",
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