zbMATH — the first resource for mathematics

Graphs and groups with tree-like properties. (English) Zbl 0696.05027
The author investigates conditions under which the Cayley graph of a finitely generated, infinite group looks like a tree. For a locally finite, infinite graph, three different notions of “metric” type are introduced to capture “tree-like”. The graph may have (1) a “uniformly spanning tree”, (2) a certain triangulation property, or (3) all its ends of finite diameter. After studying the interrelations among these properties for arbitrary graphs, the author proves that all three are equivalent for vertex-transitive graphs. A Cayley graph has one of these properties if and only if it arises from a finite extension of a free group.
Reviewer: St.Althoen

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI
[1] Cohen, D.E, Ends and free products of groups, Math. Z., 114, 9-18, (1970) · Zbl 0177.03501
[2] Dunwoody, M.J, The accessibility of finitely presented groups, Invent. math., 81, 449-457, (1985) · Zbl 0572.20025
[3] Freudenthal, H, Über die enden diskreter Räume und gruppen, Comment. math. helv., 17, 1-38, (1944) · Zbl 0060.40007
[4] Halin, R, Über unendliche wege in graphen, Math. ann., 157, 125-137, (1964) · Zbl 0125.11701
[5] Halin, R, Die maximalzahl zweiseitig unendlicher wege in graphen, Math. nachr., 44, 119-127, (1970) · Zbl 0159.25303
[6] Halin, R, Automorphisms and endomorphisms of infinite locally finite graphs, Abh. math. sem. univ. Hamburg, 39, 251-283, (1973) · Zbl 0265.05118
[7] Jung, H.A, Connectivity in infinite graphs, (), 137-143 · Zbl 0217.02603
[8] Jung, H.A, A note on fragments of infinite graphs, Combinatorica, 1, 285-288, (1981) · Zbl 0489.05036
[9] Lyndon, R.C; Schupp, P.E, ()
[10] Karrass, A; Pietrowski, A; Solitar, D, Finite and infinite cyclic extensions of free groups, J. austral. math. soc., 16, 458-466, (1973) · Zbl 0299.20024
[11] Muller, D.E; Schupp, P.E, Groups, the theory of ends, and context-free languages, J. comput. system sci., 26, 295-310, (1983) · Zbl 0537.20011
[12] Picardello, M.A; Woess, W, Martin boundaries of random walks: ends of trees and groups, Trans. amer. math. soc., 302, 185-205, (1987) · Zbl 0615.60068
[13] Picardello, M.A; Woess, W, Harmonic functions and ends of graphs, (), 457-461, (2) · Zbl 0664.60075
[14] Polat, N, Aspects topologiques de la séparation dans LES graphes infinis, Math. Z., 165, 73-100, (1979) · Zbl 0387.05007
[15] Stallings, J, ()
[16] Woess, W, Context-free languages and random walks on groups, Discrete math., 67, 81-87, (1987) · Zbl 0637.60014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.