A note on complex hyperbolic lattices and strict hyperbolization

Abstract We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney–Davis and Davis–Januszkiewicz–Weinberger. Viewing a non-uniform lattice Γ in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mtext>PU</m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \text{PU}(n,1) as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that, when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n\geq 2 , Γ is not isomorphic to any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont–Ruffoni. We also prove that a uniform lattice in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mtext>PU</m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \text{PU}(n,1) is not the fundamental group of a Charney–Davis strict hyperbolization when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n\geq 2 , assuming the initial complex satisfies some mild conditions.