On the equivalence of the definitions of volume of representations (Q901318)

Let \(\overline M\) be a compact manifold with boundary whose interior is homeomorphic to a noncompact, connected, orientable, aspherical, tame manifold \(M\), and each connected component of \(\partial\overline M\) has amenable fundamental group. For a rank-\(1\) semisimple Lie group \(G\) with trivial center and no compact factors, let \(\mathcal{X}\) be the associated symmetric space of dimension \(n=\dim M\). There are several definitions of volume of representations of \(\pi_1(M)\) into \(G\). If \(M\) is compact then for each representation \(\rho:\pi_1(M)\to G\) there is a volume of \(\rho\) defined as \(\mathrm{Vol}(\rho)=\int_Ms^*\omega_{\mathcal{X}}\) for a section \(s:M\to E_\rho\) and the Riemannian volume form \(\omega_{\mathcal{X}}\) on \(\mathcal{X}\), where \(E_\rho\) is a flat bundle over \(M\) with fiber \(\mathcal{X}\). For a noncompact manifold \(M\) the above definition of volume of representations is not valid. So far, three definitions of volume of representations have been given for noncompact manifolds. In [Geom. Dedicata 117, 111--124 (2006; Zbl 1096.51004)], \textit{S. Francaviglia} and \textit{B. Klaff} presented the definition \(D_1\) of volume of representations via a pseudodeveloping map. If \(D:\widetilde M\to\mathcal{X}\) is a pseudodeveloping map for a representation \(\rho:\pi_1(M)\to G\), that is, \(D\) is a piecewise-smooth \(\rho\)-equivariant map with the universal cover \(\widetilde M\) of \(M\), then the invariant \((D_1)\, \mathrm{Vol}_1(\rho,D)=\int_MD^*\omega_{\mathcal{X}}\) is the volume of \(\rho\). In [in: Trends in harmonic analysis. Selected papers of the conference on harmonic analysis, Rome, Italy, May 30--June 4, 2011. Berlin: Springer. 47--76 (2013; Zbl 1268.53056)], \textit{M. Bucher}, \textit{M. Burger} and \textit{A. Iozzi} used the theory of bounded cohomology to establish an invariant associated with a representation. For the pullback map \(\rho^*_b:H^n_c(G,\mathbb R)\to H^n(\pi_1(M),\mathbb R)\), the isometric isomorphism \(i^*_b:H_b^*(\overline M,\partial\overline M,\mathbb R)\to H_b^*(\overline M,\mathbb R)\) induced by the natural inclusion \(i:(\overline M,\varnothing)\to(\overline M,\partial\overline M)\), and the continuous cohomology class \([\Theta]_c\) induced by a continuous bounded cocycle \(\Theta:\mathcal{X}^{n+1}\to\mathbb R\) defined by \(\Theta(x_0,\dots,x_n)=\int_{[x_0,\dots,x_n]}\omega_{\mathcal{X}}\) with the geodesic simplex \([x_0,\dots,x_n]\), the authors define the volume as \((D_2)\, \mathrm{Vol}_2(\rho)=\langle(c\circ(i^*_b)^{-1}\circ\rho^*_b)[\Theta]_{c,b},[\overline M,\partial\overline M]\rangle\). In [Math. Z. 276, No. 3-4, 1189--1213 (2014; Zbl 1292.22003)], the author and \textit{I. Kim} gave a new definition \(D_3\) of volume of representations for a complete Riemannian manifold \(M\) with finite Lipschitz simplicial volume. If \([M]^{\ell^1}_{\mathrm{Lip}}\) is the set of all locally finite fundamental cycles of \(M\) with finite \(\ell^1\)-seminorm and finite Lipschitz constant, the authors define a third invariant \((D_3)\, \mathrm{Vol}_3(\rho)=\inf\langle\rho^*_b(\omega_b),\alpha \rangle\), where the infimum is taken over all \(\alpha\in[M]^{\ell^1}_{\mathrm{Lip}}\) and all \(\omega_b\in H^n_{c,b}(G,\mathbb R)\) with \(c(\omega_b)=\omega_{\mathcal{X}}\). In this paper, the author gives another definition \(D_4\) of volume of representations. In \(D_4\), \(\rho\)-equivariant maps are involved as in \(D_1\) and the bounded cohomology of \(M\) is involved as in \(D_2\) and \(D_3\). The invariant is defined as \((D_4)\, \mathrm{Vol}_4(\rho,D)=\langle\widehat D^*[\overline\Theta],[\widehat M]\rangle\), where \(\widehat M\) is the end compatification of \(M\) and \(\widehat D:{\widehat{\widetilde M}}\to\overline{\mathcal{X}}\) is a \(\rho\)-equivariant map that is the extension of the \(\rho\)-equivariant map \(D:\widetilde M\to \mathcal{X}\). Finally, the author shows that all the definitions are equivalent. It is proven that if \(G\) is a rank-\(1\) simple Lie group with trivial center and no compact factors, and \(M\) is a noncompact, connected, orientable, aspherical, tame manifold such that each end of \(M\) has amenable fundamental group, then the definitions \(D_1\), \(D_2\), and \(D_3\) of volume of representations of \(\pi_1(M)\) into \(G\) are equivalent. Moreover, if \(M\) admits a complete Riemannian metric with finite Lipschitz simplicial volume, then all four definitions \(D_1\), \(D_2\), \(D_3\), and \(D_4\) are equivalent.