A Conjecture Concerning a Limit of Non-Cayley Graphs
Reinhard Diestel
and Imre Leader
DOI: 10.1023/A:1011257718029
Abstract
Our aim in this note is to present a transitive graph that we conjecture is not quasi-isometric to any Cayley graph. No such graph is currently known. Our graph arises both as an abstract limit in a suitable space of graphs and in a concrete way as a subset of a product of trees.
Pages: 17–25
Keywords: Cayley graph; transitive; quasi-isometry; infinite
Full Text: PDF
References
1. L. Babai, “Vertex-transitive graphs and vertex-transitive maps,” J. Graph Theory 15 (1991), 587-627.
2. M. Gromov, “Asymptotic invariants of infinite groups,” in Geometric Group Theory, Vol. 2, G.A. Niblo and M.A. Roller (Eds.), Cambridge University Press, Cambridge, 1993.
3. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Dover, New York, 1976.
4. P.M. Soardi and W. Woess, “Amenability, unimodularity, and the spectral radius of random walks on infinite graphs,” Math. Z. 205 (1990), 471-486.
5. C. Thomassen and M.E. Watkins, “Infinite vertex-transitive, edge-transitive, non-1-transitive graphs,” Proc. Amer. Math. Soc. 105 (1989), 258-261.
6. V.I. Trofimov, “Graphs with polynomial growth,” Math. USSR-Sb. 51 (1985), 405-417.
7. W. Woess, “Topological groups and infinite graphs,” Discrete Math. 94 (1991), 1-12.
2. M. Gromov, “Asymptotic invariants of infinite groups,” in Geometric Group Theory, Vol. 2, G.A. Niblo and M.A. Roller (Eds.), Cambridge University Press, Cambridge, 1993.
3. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Dover, New York, 1976.
4. P.M. Soardi and W. Woess, “Amenability, unimodularity, and the spectral radius of random walks on infinite graphs,” Math. Z. 205 (1990), 471-486.
5. C. Thomassen and M.E. Watkins, “Infinite vertex-transitive, edge-transitive, non-1-transitive graphs,” Proc. Amer. Math. Soc. 105 (1989), 258-261.
6. V.I. Trofimov, “Graphs with polynomial growth,” Math. USSR-Sb. 51 (1985), 405-417.
7. W. Woess, “Topological groups and infinite graphs,” Discrete Math. 94 (1991), 1-12.