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In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group ''G'' has more than one end if and only if the group ''G'' admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group ''G'' has more than one end if and only if ''G'' admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions. The theorem was proved by John R. Stallings, first in the torsion-free case (1968)〔John R. Stallings. (''On torsion-free groups with infinitely many ends.'' ) Annals of Mathematics (2), vol. 88 (1968), pp. 312–334〕 and then in the general case (1971).〔John Stallings. ''Group theory and three-dimensional manifolds.'' A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.-London, 1971.〕 ==Ends of graphs== (詳細はgraph where the degree of every vertex is finite. One can view Γ as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of Γ are the ends of this topological space. A more explicit definition of the number of ends of a graph is presented below for completeness. Let ''n'' ≥ 0 be a non-negative integer. The graph Γ is said to satisfy ''e''(Γ) ≤ ''n'' if for every finite collection ''F'' of edges of Γ the graph Γ − ''F'' has at most ''n'' infinite connected components. By definition, ''e''(Γ) = ''m'' if ''e''(Γ) ≤ ''m'' and if for every 0 ≤ ''n'' < ''m'' the statement ''e''(Γ) ≤ ''n'' is false. Thus ''e''(Γ) = ''m'' if ''m'' is the smallest nonnegative integer ''n'' such that ''e''(Γ) ≤ ''n''. If there does not exist an integer ''n'' ≥ 0 such that ''e''(Γ) ≤ ''n'', put ''e''(Γ) = ∞. The number ''e''(Γ) is called ''the number of ends of'' Γ. Informally, ''e''(Γ) is the number of "connected components at infinity" of Γ. If ''e''(Γ) = ''m'' < ∞, then for any finite set ''F'' of edges of Γ there exists a finite set ''K'' of edges of Γ with ''F'' ⊆ ''K'' such that Γ − ''F'' has exactly ''m'' infinite connected components. If ''e''(Γ) = ∞, then for any finite set ''F'' of edges of Γ and for any integer ''n'' ≥ 0 there exists a finite set ''K'' of edges of Γ with ''F'' ⊆ ''K'' such that Γ − ''K'' has at least ''n'' infinite connected components. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stallings theorem about ends of groups」の詳細全文を読む スポンサード リンク
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