Integrality of volumes of representations

Abstract

Abstract Let M be an oriented complete hyperbolic n -manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation $$\rho :\pi _1(M)\rightarrow \mathrm {Isom}^+({{\mathbb {H}}}^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo>:</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>M</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>→</mml:mo><mml:msup><mml:mrow><mml:mi>Isom</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> , properly normalized, takes integer values if n is even and $$\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> . If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

Locations

• Mathematische Annalen - View - PDF