Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1995-1239799-1
MathSciNet review: 1239799
Full-text PDF

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 $ F \leq \operatorname{Aut} (X)$ could be deformed, outside some F-invariant subtree, into a uniform lattice.


References [Enhancements On Off] (What's this?)

  • [AB] R. Alperin and H. Bass, Length functions of group actions on $ \Lambda $-trees, Combinatorial Group Theory and Topology, Ann. of Math. Stud., no. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265-378. MR 895622 (89c:20057)
  • [B1] H. Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra (to appear). MR 1239551 (94j:20028)
  • [B2] -, Group actions on non-archimedean trees, Arboreal Group Theory (Roger C. Alperin, ed.), Math. Sci. Res. Inst. Publ., vol. 19, Springer-Verlag, Berlin and New York, 1988, pp. 69-131. MR 1105330 (93d:57003)
  • [BK] H. Bass and R. Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 4 (1990), 843-902. MR 1065928 (91k:20034)
  • [K] R. Kulkarni, Lattices on trees, automorphism of graphs, free groups, surfaces, preprint, CUNY, September 1988.
  • [KPS] A. Karass, A. Pietrowski, and D. Solitar, Finite and infinite cyclic extension of free groups, Austral. Math. Soc. 16 (1973), 458-466. MR 0349850 (50:2343)
  • [L1] Y. Liu, Density of the commensurability groups of uniform tree lattices, J. Algebra (to appear). MR 1273278 (95c:20036)
  • [L2] -, A necessary condition for an elliptic element to belong to a uniform tree lattice, Proc. Amer. Math. Soc. (to appear). MR 1203988 (94g:05039)
  • [L3] -, Commensurability groups of uniform tree lattices, Columbia Univ. Dissertation, 1991.
  • [Lub] A Lubotzky, Trees and discrete subgroups of Lie groups over local fields, Bull. Amer. Math. Soc. (N.S.) 20 (1988), 27-31. MR 945301 (89g:22016)
  • [S] J.-P. Serre, Trees, Springer-Verlag, New York, 1980. MR 607504 (82c:20083)

Similar Articles

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

American Mathematical Society