A counterexample to the deformation conjecture for uniform tree lattices

Liu, Ying-Sheng

doi:10.1090/S0002-9939-1995-1239799-1

A counterexample to the deformation conjecture for uniform tree lattices
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by Ying-Sheng Liu

Proc. Amer. Math. Soc. 123 (1995), 315-319

DOI: https://doi.org/10.1090/S0002-9939-1995-1239799-1

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Abstract:

Let X be a universal cover of a finite connected graph. A uniform lattice on X is a group acting discretely and cocompactly on X. We provide a counterexample to Bass and Kulkarni’s Deformation Conjecture (1990) that a discrete subgroup $F \leq \operatorname {Aut} (X)$ could be deformed, outside some F-invariant subtree, into a uniform lattice.

References

Roger Alperin and Hyman Bass, Length functions of group actions on $\Lambda$-trees, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265–378. MR 895622
Hyman Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. MR 1239551, DOI 10.1016/0022-4049(93)90085-8
Hyman Bass, Group actions on non-Archimedean trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 69–131. MR 1105330, DOI 10.1007/978-1-4612-3142-4_{3}
Hyman Bass and Ravi Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 3 (1990), no. 4, 843–902. MR 1065928, DOI 10.1090/S0894-0347-1990-1065928-2

Lattices on trees, automorphism of graphs, free groups, surfaces

A. Karrass, A. Pietrowski, and D. Solitar, Finite and infinite cyclic extensions of free groups, J. Austral. Math. Soc. 16 (1973), 458–466. Collection of articles dedicated to the memory of Hanna Neumann, IV. MR 0349850
Ying-Sheng Liu, Density of the commensurability groups of uniform tree lattices, J. Algebra 165 (1994), no. 2, 346–359. MR 1273278, DOI 10.1006/jabr.1994.1115
Ying-Sheng Liu, A necessary condition for an elliptic element to belong to a uniform tree lattice, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1035–1039. MR 1203988, DOI 10.1090/S0002-9939-1994-1203988-1

Commensurability groups of uniform tree lattices

Alexander Lubotzky, Trees and discrete subgroups of Lie groups over local fields, Bull. Amer. Math. Soc. (N.S.) 20 (1989), no. 1, 27–30. MR 945301, DOI 10.1090/S0273-0979-1989-15686-3
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504