A counterexample to the deformation conjecture for uniform tree lattices

Author:
Ying-Sheng Liu

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

MSC:
Primary 20E08; Secondary 05C25, 20F32

MathSciNet review:
1239799

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 could be deformed, outside some *F*-invariant subtree, into a uniform lattice.

**[AB]**Roger Alperin and Hyman Bass,*Length functions of group actions on Λ-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****[B1]**Hyman Bass,*Covering theory for graphs of groups*, J. Pure Appl. Algebra**89**(1993), no. 1-2, 3–47. MR**1239551**, 10.1016/0022-4049(93)90085-8**[B2]**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**, 10.1007/978-1-4612-3142-4_3**[BK]**Hyman Bass and Ravi Kulkarni,*Uniform tree lattices*, J. Amer. Math. Soc.**3**(1990), no. 4, 843–902. MR**1065928**, 10.1090/S0894-0347-1990-1065928-2**[K]**R. Kulkarni,*Lattices on trees, automorphism of graphs, free groups, surfaces*, preprint, CUNY, September 1988.**[KPS]**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****[L1]**Ying-Sheng Liu,*Density of the commensurability groups of uniform tree lattices*, J. Algebra**165**(1994), no. 2, 346–359. MR**1273278**, 10.1006/jabr.1994.1115**[L2]**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**, 10.1090/S0002-9939-1994-1203988-1**[L3]**-,*Commensurability groups of uniform tree lattices*, Columbia Univ. Dissertation, 1991.**[Lub]**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**, 10.1090/S0273-0979-1989-15686-3**[S]**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20E08,
05C25,
20F32

Retrieve articles in all journals with MSC: 20E08, 05C25, 20F32

Additional Information

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

Article copyright:
© Copyright 1995
American Mathematical Society