The second bounded cohomology

of an amalgamated free product of groups

Author:
Koji Fujiwara

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1113-1129

MSC (1991):
Primary 20F32; Secondary 55U99, 20E06.

DOI:
https://doi.org/10.1090/S0002-9947-99-02282-5

Published electronically:
July 7, 1999

MathSciNet review:
1491864

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite dimensional second bounded cohomology.

**[BaGh1]**J. Barge, E. Ghys, Surfaces et cohomologie bornée, Invent. Math., 92, 1988, 509-526. MR**89e:55015****[BaGh2]**J. Barge, E. Ghys, Cocycles bornés et actions de groupes sur les arbres réels, in ``Group Theory from a Geometric Viewpoint'', World Sci. Pub., New Jersey, 1991, 617-622. MR**93f:20033****[BaGh3]**J. Barge, E. Ghys, Cocycles d'Euler et de Maslov, Math. Ann., 294, no 2, 1992, 235-265. MR**95b:55021****[Bav]**C. Bavard, Longueur stable des commutateurs, L'Enseignement Math., 37, 1991, 109-150. MR**92g:20051****[B]**R. Brooks, Some remarks on bounded cohomology, Ann. Math. Studies, 97, 1981, 53-63. MR**83a:57038****[BS]**R. Brooks, C. Series, Bounded cohomology for surface groups, Topology, 23, no 1, 1984, 29-36. MR**85c:57009****[Br]**K.S. Brown, ``Cohomology of groups'', Springer, New York, 1982. MR**83k:20002****[EF]**D.B.A. Epstein, K. Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36, 1997, 1275-1289. MR**98k:20088****[F]**K. Fujiwara, The second bounded cohomology of a group acting on a Gromov-hyperbolic space, Proc. London Math Soc. (3) 76, 1998, 70-94. MR**99c:20072****[FO]**K. Fujiwara, K. Ohshika, The second bounded cohomology of 3-manifolds, preprint, Univ Tokyo, 1997.**[Gh]**E. Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée, Contemp. Math. 58, AMS, 1987, 81-106. MR**88m:58024****[Gr]**R.I. Grigorchuk, Some remarks on bounded cohomology, LMS Lecture Note 204, Cambridge Univ Press, Cambridge, 1995, 111-163. MR**96j:20073****[G]**M. Gromov, Volume and bounded cohomology, Publ. Math. IHES, 56, 1982, 5-99. MR**84h:53053****[I]**N.V. Ivanov, Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 143, 1985, 69-109. MR**87b:53070****[LS]**R.C. Lyndon, P.E. Schupp, ``Combinatorial Group Theory'', Berlin, Springer, 1977. MR**58:28182****[MaMo]**S. Matsumoto, S. Morita, Bounded cohomology of certain groups of homomorphisms, Proc. A.M.S., 94, 1985, 539-544. MR**87e:55006****[Mi]**Y. Mitsumatsu, Bounded cohomology and -homology of surfaces, Topology, 23, 1984, 465-471. MR**86f:57010****[So1]**T. Soma, Bounded cohomology of closed surfaces, Topology 36, 1997, 1221-1246. MR**99a:57011****[So2]**T. Soma, Bounded cohomology and topologically tame Kleinian groups, Duke Math Jour. 88, 1997, no 2, 357-370. MR**98f:57023****[S]**J. Stallings, ``Group theory and three-dimensional manifolds'', Yale Univ. Press, New Haven, 1971. MR**54:3705****[Y]**T. Yoshida, On 3-dimensional bounded cohomology of surfaces, in ``Homotopy Theory and Related Topics'', Advanced Studies in Pure Math 9, Kinokuniya, Tokyo, 1986, 173-176. MR**88m:57032**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
20F32,
55U99,
20E06.

Retrieve articles in all journals with MSC (1991): 20F32, 55U99, 20E06.

Additional Information

**Koji Fujiwara**

Affiliation:
Department of Mathematics, Keio University, Yokohama, 223 Japan

Address at time of publication:
Math Institute, Tohoku Univeristy, Sendai, 980-8578, Japan

Email:
fujiwara@math.tohoku.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-99-02282-5

Keywords:
Bounded cohomology,
ends of groups

Received by editor(s):
February 20, 1996

Received by editor(s) in revised form:
November 7, 1997

Published electronically:
July 7, 1999

Additional Notes:
Most of the work was done when the author visited MSRI supported in part by NSF grant DMS-9022140 and a JSPS grant. He is supported in part by The Inamori Foundation.

Dedicated:
Dedicated to Professor John Stallings for his 60th birthday

Article copyright:
© Copyright 1999
American Mathematical Society