Graph of groups

<p>In <a href="/facts/Geometric_group_theory/AJhxpFUe">geometric group theory</a>, a graph of groups is an object consisting of a collection of <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a> indexed by the vertices and edges of a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a>, together with a family of <a href="/facts/Monomorphism/Q9BrtDTs">monomorphisms</a> of the edge groups into the vertex groups.
There is a unique group, called the fundamental group, canonically associated to each finite <a href="/facts/Connectivity_(graph_theory)/FSNjsYhz">connected</a> graph of groups. It admits an orientation-preserving <a href="/facts/Group_action/Vh9VtUUw">action</a> on a <a href="/facts/Tree_(graph_theory)/TCWk4Joy">tree</a>: the original graph of groups can be recovered from the <a href="/facts/Quotient_graph/EpJlV5UV">quotient graph</a> and the <a href="/facts/Stabilizer_subgroup/Vh9VtUUw">stabilizer subgroups</a>. This theory, commonly referred to as <a href="/facts/Bass%25E2%2580%2593Serre_theory/qCI1VD4g">Bass–Serre theory</a>, is due to the work of <a href="/facts/Hyman_Bass/Se6dGEd6">Hyman Bass</a> and <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Jean-Pierre Serre</a>.
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Graph of groups open-in-new

Graph of groups