Homological and finiteness properties of picture groups
HTML articles powered by AMS MathViewer
- by Daniel S. Farley PDF
- Trans. Amer. Math. Soc. 357 (2005), 3567-3584 Request permission
Abstract:
Picture groups are a class of groups introduced by Guba and Sapir. Known examples include Thompson’s groups $F$, $T$, and $V$. In this paper, a large class of picture groups is proved to be of type $F_{\infty }$. A Morse-theoretic argument shows that, for a given picture group, the rational homology vanishes in almost all dimensions.References
- Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. MR 1465330, DOI 10.1007/s002220050168
- R. L. Graham, M. Grötschel, and L. Lovász (eds.), Handbook of combinatorics. Vol. 1, 2, Elsevier Science B.V., Amsterdam; MIT Press, Cambridge, MA, 1995. MR 1373655
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- Kenneth S. Brown, Finiteness properties of groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 45–75. MR 885095, DOI 10.1016/0022-4049(87)90015-6
- Kenneth S. Brown, The geometry of finitely presented infinite simple groups, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 121–136. MR 1230631, DOI 10.1007/978-1-4613-9730-4_{5}
- Kenneth S. Brown, The geometry of rewriting systems: a proof of the Anick-Groves-Squier theorem, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 137–163. MR 1230632, DOI 10.1007/978-1-4613-9730-4_{6}
- Kenneth S. Brown and Ross Geoghegan, An infinite-dimensional torsion-free $\textrm {FP}_{\infty }$ group, Invent. Math. 77 (1984), no. 2, 367–381. MR 752825, DOI 10.1007/BF01388451
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438
- Farley, D. S. Actions of Picture Groups on CAT(0) Cubical Complexes to appear in Geometriae Dedicata.
- Daniel S. Farley, Finiteness and $\rm CAT(0)$ properties of diagram groups, Topology 42 (2003), no. 5, 1065–1082. MR 1978047, DOI 10.1016/S0040-9383(02)00029-0
- Daniel S. Farley, Proper isometric actions of Thompson’s groups on Hilbert space, Int. Math. Res. Not. 45 (2003), 2409–2414. MR 2006480, DOI 10.1155/S107379280321014X
- Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145. MR 1612391, DOI 10.1006/aima.1997.1650
- Marvin J. Greenberg and John R. Harper, Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. MR 643101
- Victor Guba and Mark Sapir, Diagram groups, Mem. Amer. Math. Soc. 130 (1997), no. 620, viii+117. MR 1396957, DOI 10.1090/memo/0620
- Guba, V.S.; Sapir, M.V. Diagram groups and directed 2-complexes: homotopy and homology. preprint at front.math.ucdavis.edu.
- V. S. Guba and M. V. Sapir, On subgroups of the R. Thompson group $F$ and other diagram groups, Mat. Sb. 190 (1999), no. 8, 3–60 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 7-8, 1077–1130. MR 1725439, DOI 10.1070/SM1999v190n08ABEH000419
- V. S. Guba and M. V. Sapir, Rigidity properties of diagram groups, Internat. J. Algebra Comput. 12 (2002), no. 1-2, 9–17. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). MR 1902358, DOI 10.1142/S0218196702000882
- Vesna Kilibarda, On the algebra of semigroup diagrams, Internat. J. Algebra Comput. 7 (1997), no. 3, 313–338. MR 1448329, DOI 10.1142/S0218196797000150
- Ralph McKenzie and Richard J. Thompson, An elementary construction of unsolvable word problems in group theory, Word problems: decision problems and the Burnside problem in group theory (Conf., Univ. California, Irvine, Calif. 1969; dedicated to Hanna Neumann), Studies in Logic and the Foundations of Math., vol. 71, North-Holland, Amsterdam, 1973, pp. 457–478. MR 0396769, DOI 10.1016/0003-4916(72)90140-6
- Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Additional Information
- Daniel S. Farley
- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- Received by editor(s): December 4, 2003
- Published electronically: December 9, 2004
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 3567-3584
- MSC (2000): Primary 20J05, 20F65
- DOI: https://doi.org/10.1090/S0002-9947-04-03720-1
- MathSciNet review: 2146639