2006/03/31 by Alexandru Dimca, Ştefan Papadima, Stefan Papadima +1 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AG #math.GR #msc:14F35 #msc:14M12 #msc:20F36 #msc:57M07
paper · pdf · doi:10.1090/s1056-3911-07-00463-8
published as Journal of Algebraic Geometry 17 (2008), no. 1, 185-197 · 11 pages, accepted for publication by the Journal of Algebraic Geometry
arxiv created 2006/10/14 · openalex publication_date 2007/06/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A finite simple graph <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">Γ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> determines a right-angled Artin group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript normal upper Gamma"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mi mathvariant="normal"> Γ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">GΓ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with one generator for each vertex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and with one commutator relation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v w equals w v"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mi>w</mml:mi> <mml:mo>=</mml:mo> <mml:mi>w</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">vw=wv</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each pair of vertices joined by an edge. The Bestvina-Brady group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript normal upper Gamma"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mi mathvariant="normal"> Γ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">NΓ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the kernel of the projection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript normal upper Gamma Baseline right-arrow double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi mathvariant="normal"> Γ </mml:mi> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">GΓ → \mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which sends each generator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We establish precisely which graphs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">Γ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> give rise to quasi-Kähler (respectively, Kähler) groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript normal upper Gamma"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mi mathvariant="normal"> Γ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">NΓ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasi-projective variety.