Quasi-isometric invariance of continuous group \(L^p\)-cohomology, and first applications to vanishings (Q2220246)

Let \(G\) be a locally compact second countable group, e.g.\ a Lie group. For \(p>1\), its group \(L^p\)-cohomology is defined as the continuous group cohomology of \(G\) with coefficients in the right-regular representation of \(G\) on \(L^p(G)\). (The latter is defined with respect to left-invariant Haar measure.) If \(G\) acts properly discontinuously, freely and cocompactly, by simplicial automorphisms on a simplicial complex \(X\), then the group \(L^p\)-cohomology of \(G\) agrees with the simplicial \(L^p\)-cohomology of \(X\). (This is proved in Proposition 3.2 of the paper under review.) However, the focus of the paper under review is on non-discrete groups. In this general case, the authors prove in Section 4 that the group \(L^p\)-cohomology of \(G\) agrees with the asymptotic \(L^p\)-cohomology of \(G\). The latter was defined by P. Pansu [1995, unpublished], its defining \(k\)-th cochain group consists of functions \(u\colon G^{k+1}\to{\mathbb R}\) satisfying a certain finiteness condition: for each \(s>0\) and \(\Delta_s^{(k)}=\left\{(g_0,\ldots,g_k)\colon d(g_i,g_j)\le s\right\}\) one demands \(\int_{\Delta_s^{(k)}}\vert u(g_0,\ldots,g_k)\vert^p<\infty\). Pansu proved that his asymptotic \(L^p\)-cohomology is invariant under quasi-isometries. The authors thus obtain that group \(L^p\)-cohomology is invariant under quasi-isometries, the main result of this paper. Quasi-isometric invariance implies that the group \(L^p\)-cohomology of a simple Lie group is isomorphic to the group \(L^p\)-cohomology of any of its parabolic subgroups \(P\). Such a \(P\) is a semidirect product \(P=M\ltimes AN\), where \(M\) is semi-simple and \(AN\) is the solvable radical of \(P\). For \(G=\mathrm{SL}(n,{\mathbb R})\) one has that \(AN\) is quasi-isometric to hyperbolic \(d\)-space (with \(d\) depending on the choice of \(P\)). The simplicial \(L^p\)-cohomology of hyperbolic space has been computed by Pansu, and the group \(L^p\)-cohomology of \(AN\) is isomorphic to that of any cocompact lattice and thus to the simplicial \(L^p\)-cohomology of hyperbolic \(d\)-space. From this, the authors obtain the group \(L^p\)-cohomology of \(AN\) and then they develop an analogue of the Hochschild-Serre spectral sequence to compute the group \(L^p\)-cohomology of \(P=M\ltimes AN\) and thus of \(G\). The result is that the \(k\)-th group \(L^p\)-cohomology of \(G=\mathrm{SL}(n,{\mathbb R})\) vanishes for \(k\le \left[\frac{n^2}{4}\right]\cdot\frac{1}{p}\) and \(k\ge\left[\frac{n^2}{4}\right]\cdot\frac{1}{p}+\left[\frac{n^2+1}{4}\right]\). A similar line of argument works whenever a simple Lie group \(G\) has a parabolic subgroup \(P\), whose solvable radical \(AN\) is quasi-isometric to hyperbolic \(d\)-space for some \(d\). The authors call such Lie groups ``admissible'' and they give in section 7 a classification of admissible simple Lie groups: the admissible simple Lie groups are those whose relative root system is of type \(A_l, B_l, C_l, D_l, E_6\) or \(E_7\). Thus, for classical types the only excluded groups are those with non-reduced root system \(BC_l\). The general result, which the authors then obtain for admissible simple Lie groups \(G\), is that the \(k\)-th group \(L^p\)-cohomology of \(G\) vanishes for \(k\le \frac{d-1}{p}\) or \(k\ge \frac{d-1}{p}-d+2+\dim(G/K)\).