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  "body": "\n## Statement\n\nLet $G$ be a [[finitely generated group|082DWWXMRJ]] and let $\\Gamma$ be its [[Cayley graph|0GYQK9NDJF]] for a finite generating set.\nFor a finite set $K$ of vertices and edges of $\\Gamma$, count the infinite connected components of $\\Gamma \\setminus K$.\nThe number of ends $e(G)$ is the supremum of this count over all such $K$.\nIt does not depend on the generating set.\nThen $e(G) \u003e 1$ if and only if one of the following holds:\n\n1. $G$ is an amalgamated free product $A *_C B$ with $C$ finite, $C \\neq A$ and $C \\neq B$.\n2. $G$ is an [[HNN extension|YEYERRY37K]] $A *_C$ with $C$ a finite [[subgroup|0X1JXT47GA]] of $A$.\n\n## Remarks\n\nThe number of ends of a finitely generated group is always $0$, $1$, $2$ or $\\infty$.\nIt is $0$ exactly for finite groups and $2$ exactly for infinite groups with an infinite cyclic subgroup of finite index.\n\nThe easy direction: if $G$ splits over a finite subgroup, then $G$ acts on the Bass–Serre tree of the splitting.\nRemoving a single edge of this tree separates the Cayley graph into at least two infinite parts.\nThe hard direction, due to Stallings, builds from a set of vertices with finite boundary in $\\Gamma$ a tree on which $G$ acts with finite edge stabilisers.\n\nExample: $\\Z$ has two ends, since removing one vertex of the line leaves two rays; and $\\Z$ is the HNN extension of the trivial group over the trivial subgroup.\nExample: the [[free group|0KTW26C80J]] $F_2 = \\Z * \\Z$ is an amalgam over the trivial group, and it has infinitely many ends.\nExample: $\\Z^2$ has one end, since the complement of a finite set in the grid has one infinite component; so $\\Z^2$ does not split over a finite subgroup.\n\n## Explanations\n\n### L1 human\n\nRemove a big finite chunk from the Cayley graph of a group and count the infinite pieces that remain; the largest such count is the number of ends.\nThe line has two ends, the plane has one.\nThe theorem says that a group has more than one end exactly when it can be cut, algebraically, into two pieces glued along a finite group.\nSo a picture at infinity detects an algebraic splitting.\n\n### L2 human\n\nThe theorem is a cornerstone of [[Bass–Serre theory|6M731X66RD]] and of [[geometric group theory|2T3218RXTD]].\nSince the number of ends is a [[quasi-isometry|6FKYSH729T]] invariant, splitting over a finite subgroup is a quasi-isometry invariant property.\nIt implies that a finitely generated torsion-free group with a free subgroup of finite index is itself free.\nIterating splittings over finite groups leads to the accessibility theorem of Dunwoody for finitely presented groups.\n",
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