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SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS

JOHN CRISP

I.M.B. (UMR 5584 du CNRS), Université de Bourgogne, B.P. 47 870, 21078 Dijon, France

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MICHAH SAGEEV

Technion, Israel University of Technology, Department of Mathematics, Haifa 32000, Israel

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, and

MARK SAPIR

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

The research of the third author was supported in part by NSF and BSF (the US-Israeli) grants.

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https://doi.org/10.1142/S0218196708004536Cited by:14 (Source: Crossref)

Abstract

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight "forbidden" graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right-angled Artin group to contain no hyperbolic surface subgroups.

We prove that for one of these "forbidden" subgraphs P₂(6), the right-angled Artin group A(P₂(6)) is a subgroup of a (right-angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).

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