zbMATH — the first resource for mathematics

Groups of piecewise linear homeomorphisms of the real line. (English) Zbl 0563.57022
The paper studies the group PLF(\({\mathbb{R}})\) of orientation-preserving homeomorphisms of the real line \({\mathbb{R}}\) which are piecewise-linear with respect to a finite subdivision of \({\mathbb{R}}\). The main results are presentations of PLF(\({\mathbb{R}})\) and certain of its subgroups; a proof that PLF(\({\mathbb{R}})\) contains no free subgroups of rank greater than 1; and a proof that PLF(\({\mathbb{R}})\) satisfies no laws. Also constructed is a sequence of finitely-presented subgroups G(p) of PLF(\({\mathbb{R}})\) which contain no free subgroups of rank greater than 1 and satisfy no laws. (R. Geoghegan has suggested the possibility that among the G(p)’s there might be a finitely presented counterexample to a conjecture of von Neumann: A discrete group is non-amenable if and only if it contains a subgroup isomorphic to the free group of rank 2.)

57S25 Groups acting on specific manifolds
20F05 Generators, relations, and presentations of groups
20F60 Ordered groups (group-theoretic aspects)
57Q99 PL-topology
Full Text: DOI EuDML
[1] [A] Adjan, S.I.: Random walks on free periodic groups. Math. USSR Izvestiya21, 425-434 (1983) · Zbl 0528.60011 · doi:10.1070/IM1983v021n03ABEH001799
[2] [B] Bourbaki, N.: Groupes et algèbres de Lie, IV?VI, Éléments de math. Fasc. XXXIV. Paris: Hermann 1968 · Zbl 0186.33001
[3] [BG] Brown, K.S., Geoghegan, R.: An infinite-dimensional torsion-freeFP ? group. Invent. Math.77, 367-381 (1984) · Zbl 0557.55009 · doi:10.1007/BF01388451
[4] [D] Dydak, J.: A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR’s. Bull. Acad. Pol. Sci. (Ser. Sci. Math. Astr. Phys.)25, 55-62 (1977) · Zbl 0357.55018
[5] [E] Epstein, D.B.A.: The simplicity of certain groups of homeomorphisms. Compositio Math. period22, 165-173 (1970 · Zbl 0205.28201
[6] [F] Freyd, P.: Letter to A. Heller. (4/11/81)
[7] [FH] Freyd, P., Heller, A.: Splitting homotopy idempotents, II. Mimeographed, U. Penn., 1979 · Zbl 0786.55008
[8] [FK] Fricke, R., Klein, F.: Vorlesungen über die Theorie der Automorphen Funktionen, vol. 1. Leipzig: Teubner 1897
[9] [G] Glass, A.M.W.: Ordered permutation groups. London. Math. Soc. Lect. Note Ser. 55. Cambridge: Cambridge U. Press 1981 · Zbl 0473.06010
[10] [H] Higman, G.: Finitely presented infinite simple groups. Notes in Pure Math. 8, Australian Nat. U., Canberra, 1974
[11] [L] Lyndon, R.C.: Cohomology theory of groups with a single defining relation. Ann. of Math.52, 650-665 (1950) · Zbl 0039.02302 · doi:10.2307/1969440
[12] [M] Macbeath, A.M.: Groups of homeomorphisms of a simply connected space. Ann. of Math.79, 473-488 (1964) · Zbl 0122.17503 · doi:10.2307/1970405
[13] [MT] McKenzie, R., Thompson, R.J.: An elementary construction of unsolvable problems in group theory, Word Problems, Boone, W.W., Cannonito, F.B., Lyndon, R.C. (eds). Amsterdam: North Holland Publishing Company 1973, pp. 457-478
[14] [MKS] Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. 2nd ed., New York: Dover 1976 · Zbl 0362.20023
[15] [O] Ol’shanskii, A.Yu.: On the question of existence of an invariant mean on a group. Russian Math. Surveys35(4), 180-181 (1980) · Zbl 0465.20030 · doi:10.1070/RM1980v035n04ABEH001876
[16] [P] Passman, D.S.: The algebraic structure of group rings. New York: John Wiley & Sons 1977 · Zbl 0368.16003
[17] [T] Thompson, R.J.: Embeddings into finitely generated simple groups which preserve the word problem, Word Problems II, Adjan, S.I., Boone, W.W., Higman, G. (eds). Amsterdam: North Holland Publishing Company 1980, pp. 401-441
[18] [VN] von Neumann, J.: Zur allgemeinen Theorie des Masses. Fund. Math.13, 73-116 (1929) · JFM 55.0151.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.