[Paper Review] Proportionality principle for the simplicial volume of families of Q-rank 1 locally symmetric spaces

Michelle Bucher, Инканг Ким+1 more|arXiv (Cornell University)|Sep 21, 2012

Advanced Algebra and Geometry20 references4 citations

TL;DR

This paper establishes the proportionality principle between the Riemannian volume and the locally finite simplicial volume for Q-rank 1 locally symmetric spaces whose universal covers are products of real hyperbolic spaces, even when cusp groups are non-amenable. Using bounded cohomology and the comparison map between continuous bounded and ordinary cohomology, the authors prove that the simplicial volume equals the volume divided by the sup norm of the volume form on the universal cover, extending known results beyond amenable cusp groups.

ABSTRACT

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of Q-rank 1 locally symmetric spaces with amenable cusp groups established by Löh and Sauer.

Motivation & Objective

• To extend the proportionality principle for simplicial volume beyond amenable cusp groups in Q-rank 1 locally symmetric spaces.
• To establish the equality between locally finite simplicial volume and Riemannian volume divided by the sup norm of the volume form on the universal cover.
• To provide a self-contained proof using bounded cohomology techniques, avoiding reliance on measure homology or multicomplexes.
• To demonstrate that the proportionality principle holds for reducible Q-rank 1 locally symmetric spaces, where cusp fundamental groups are non-amenable.
• To confirm the validity of the proportionality principle in cases where previous methods based on amenability fail.

Proposed method

• The authors define a bounded $H$-invariant cocycle $\Theta$ on the universal cover $\widetilde{M}$ using integration of the volume form $\omega_{\widetilde{M}}$ over geodesic simplices.
• They show that $\Theta$ determines a continuous bounded cohomology class $[\Theta]^{H}_{b}$ in $H^{n}_{c,b}(H,\mathbb{R}_{\varepsilon})$, which maps to the volume class $\omega_{\widetilde{M}}$ under the comparison map.
• By assuming the comparison map $c: H^{n}_{c,b}(H,\mathbb{R}_{\varepsilon}) \to H^{n}_{c}(H,\mathbb{R}_{\varepsilon})$ is an isomorphism, they relate the bounded cohomology class to the simplicial volume via duality.
• They use the geodesic straightening map on locally finite chains to relate the $\ell^1$-homology class of the fundamental cycle to the volume via pairing with $\Theta$, showing that the pairing yields $\mathrm{Vol}(M)$.
• They derive the inequality $\|M\|_{\mathrm{lf}} \geq \mathrm{Vol}(M)/\|\omega_{\widetilde{M}}\|_{\infty}$ and prove equality via a reverse inequality from the bounded cohomology framework.
• The proof relies on isometric embeddings of cohomology groups induced by inclusions of the identity component $G$ of $H$ and the lattice $\Gamma$, preserving norms.

Experimental results

Research questions

• RQ1 Does the proportionality principle between Riemannian volume and locally finite simplicial volume hold for Q-rank 1 locally symmetric spaces with non-amenable cusp groups?
• RQ2 Can the proportionality principle be established using bounded cohomology techniques independent of measure homology or Gromov's multicomplexes?
• RQ3 Is the comparison map between continuous bounded cohomology and continuous cohomology an isomorphism in top degree for products of real hyperbolic spaces?
• RQ4 Does the simplicial volume of a Q-rank 1 locally symmetric space covered by a product of R-rank 1 symmetric spaces equal $\mathrm{Vol}(M)/\|\omega_{\widetilde{M}}\|_{\infty}$?
• RQ5 Can the proportionality principle be extended to reducible Q-rank 1 locally symmetric spaces, where cusp groups are non-amenable?

Key findings

• The proportionality principle holds for Q-rank 1 locally symmetric spaces whose universal cover is a product of real hyperbolic spaces $\mathbb{H}^{n_1} \times \cdots \times \mathbb{H}^{n_k}$ with $n_i \geq 2$, even when cusp groups are non-amenable.
• The locally finite simplicial volume satisfies $\|M\|_{\mathrm{lf}} = \mathrm{Vol}(M)/\|\omega_{\widetilde{M}}\|_{\infty}$ under the assumption that the comparison map $c: H^{n}_{c,b}(H,\mathbb{R}_{\varepsilon}) \to H^{n}_{c}(H,\mathbb{R}_{\varepsilon})$ is an isomorphism.
• The proof provides a direct, self-contained argument using bounded cohomology, avoiding the discrete approximation techniques of Löh and Sauer.
• The equality $\|\omega_M\|^\infty = \|\omega_{\widetilde{M}}\|_{\infty}$ is established via norm preservation under the chain of cohomological restrictions and the comparison map.
• The result confirms that the proportionality principle extends to reducible Q-rank 1 locally symmetric spaces, where cusp fundamental groups are non-amenable.
• The work supports the conjecture that the proportionality principle holds for all Q-rank 1 locally symmetric spaces, regardless of cusp group amenability.