Surface mapping class groups are ultrahopfian

Date

2000-07-01

Author

Korkmaz, Mustafa

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

31
views

0
downloads

Let S denote a compact, connected, orientable surface with genus g and h boundary components. We refer to S as a surface of genus g with h holes. Let [Mscr ]S denote the mapping class group of S, the group of isotopy classes of orientation-preserving homeomorphisms S → S. Let G be a group. G is hopfian if every homomorphism from G onto itself is an automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists a normal subgroup of finite index in G which does not contain g. Every finitely generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian ([2, 3]) if every homomorphism ψ G → G with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots. By a result of Grossman [5], [Mscr ]S is residually finite. Since [Mscr ]S is also finitely generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian. In this paper, we shall answer a more general question. We say that a group G is ultrahopfian if every homomorphism ψ: G → G with ψ(G) normal in G and G/ψ(G) abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We shall prove the following result.

URI

https://hdl.handle.net/11511/47648

Journal

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY

DOI

https://doi.org/10.1017/s0305004199004259

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

Arbitrarily long factorizations in mapping class groups Korkmaz, Mustafa (2015-05-12) On a compact oriented surface of genus g with n ≥ 1 boundary components, δ1, δ2, ..., δn, we consider positive factorizations of the boundary multitwist tδ1 tδ2 ...tδn, where tδi is the positive Dehn twist about the boundary δi. We prove that for g ≥ 3, the boundary multitwist tδ1 tδ2 can be written as a product of arbitrarily large number of positive Dehn twists about nonseparating simple closed curves, extending a recent result of Baykur and Van Horn-Morris, who proved this result for g ≥ 8. This fact has...
Automorphisms of curve complexes on nonorientable surfaces Atalan, Ferihe; Korkmaz, Mustafa (2014-01-01) For a compact connected nonorientable surface N of genus g with n boundary components, we prove that the natural map from the mapping class group of N to the automorphism group of the curve complex of N is an isomorphism provided that g + n >= 5. We also prove that two curve complexes are isomorphic if and only if the underlying surfaces are diffeomorphic.
Arbitrarily Long Factorizations in Mapping Class Groups DALYAN, ELİF; Korkmaz, Mustafa; Pamuk, Mehmetcik (2015-01-01) On a compact oriented surface of genus g with n= 1 boundary components, d1, d2,..., dn, we consider positive factorizations of the boundary multitwist td1 td2 tdn, where tdi is the positive Dehn twist about the boundary di. We prove that for g= 3, the boundary multitwist td1 td2 can be written as a product of arbitrarily large number of positive Dehn twists about nonseparating simple closed curves, extending a recent result of Baykur and Van Horn- Morris, who proved this result for g= 8. This fact has immed...
Automorphisms of the Hatcher-Thurston complex Irmak, Elmas; Korkmaz, Mustafa (Springer Science and Business Media LLC, 2007-12-01) Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Ant HT(S) is isomorphic to the extended mapping class group of S modulo its center.
Automorphisms of complexes of curves on odd genus nonorientable surfaces Atalan Ozan, Ferihe; Korkmaz, Mustafa; Department of Mathematics (2005) Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n > 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group M of N.

Citation Formats

M. Korkmaz, “Surface mapping class groups are ultrahopfian,” MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, pp. 35–53, 2000, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/47648.