Set of stallings

The set of Stallings is a theorem from the mathematical field of group theory characterized, the groups having more than one end. This gives rise to the characterization of free groups through their cohomological dimension , which is sometimes also referred to as the Stallings' set or Stallings-Swan's set .

John Stallings and Richard Swan received the Cole Prize for Algebra for this .

Set of stallings across ends of groups

For a finitely generated group denote the number of ends of the Cayley graph of . (This number is independent of the choice of the generating system used for the construction of the Cayley graph.) According to a Freudenthal theorem, either or holds . ${\displaystyle G}$ ${\displaystyle e(G)}$ ${\displaystyle G}$ ${\displaystyle e(G)\leq 2}$ ${\displaystyle e(G)=\infty }$

The set of Stallings says that is the case if and when ${\displaystyle e(G)>1}$ ${\displaystyle G}$

• either as a nontrivial amalgamated product of two finitely generated groups over a finite amalgamated subgroup ${\displaystyle G=G_{1}*_{A}G_{2}}$
• or as a nontrivial HNN extension of a finitely generated group over a finite group ${\displaystyle G=G_{1}*_{A}}$

can be disassembled.

In particular, for torsion-free finitely generated groups if and only if there is a free product of two nontrivial subgroups. ${\displaystyle e(G)=\infty }$ ${\displaystyle G}$ ${\displaystyle G=G_{1}*G_{2}}$

Stallings-Swan's theorem on characterization of free groups

From Stallings' theorem it follows that a finitely generated group is free if and only if holds for its cohomological dimension . ${\displaystyle cd_{\mathbb {Z} }(G)=1}$

A more general form was proven by Swan. Let be a ring with one and a torsion-free group. Then is free if and only if applies. ${\displaystyle R}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle cd_{R}(G)=1}$This theorem does without the assumption that is finitely generated. The assumption of freedom from torsion is always fulfilled for groups . ${\displaystyle G}$ ${\displaystyle cd_{\mathbb {Z} }(G)<\infty }$

A further implication is that a torsion-free group containing a free subgroup of finite index must itself be free.

literature

• John Stallings: On torsion-free groups with infinitely many ends. Ann. of Math. (2) 88: 312-334 (1968).
• Richard Swan: Groups of cohomological dimension one. J. Algebra 12: 585-610 (1969).
• Daniel Cohen: Groups of cohomological dimension one. Lecture Notes in Mathematics, Vol. 245 (1972), Springer-Verlag, Berlin-New York.
• Martin Dunwoody: Accessibility and groups of cohomological dimension one. Proc. London Math. Soc. (3) 38 (1979) no. 2, 193-215.
• Martin Dunwoody: The accessibility of finitely presented groups. Invent. Math. 81 (1985) no. 3, 449-457.
• Michail Gromow: Hyperbolic groups. Essays in group theory, Math. Sci. Res. Inst. Publ. 8 (1987) Springer, New York, 75-263.
• Graham Niblo: A geometric proof of Stallings' theorem on groups with more than one end. Geom. Dedicata 105: 61-76 (2004).
• Michail Kapovich: Energy of harmonic functions and Gromov's proof of Stallings' theorem. Georgian Math. J. 21 (2014), no. 3, 281-296.