Geometry & Topology Volume 29, issue 9 (2025)

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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$L^p$-cohomology for higher-rank spaces and groups: critical exponents and rigidity

Marc Bourdon and Bertrand Rémy

Geometry & Topology 29 (2025) 4799–4839

DOI: 10.2140/gt.2025.29.4799

Bibliography

1	R Azencott, E N Wilson, Homogeneous manifolds with negative curvature, I, Trans. Amer. Math. Soc. 215 (1976) 323 MR394507
2	A Borel, The L²-cohomology of negatively curved Riemannian symmetric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985) 95 MR802471
3	M Bourdon, B Rémy, Quasi-isometric invariance of continuous group L^p-cohomology, and first applications to vanishings, Ann. H. Lebesgue 3 (2020) 1291 MR4191391
4	M Bourdon, B Rémy, Non-vanishing for group L^p-cohomology of solvable and semisimple Lie groups, J. Éc. polytech. Math. 10 (2023) 771 MR4585549
5	R Bryant, P Griffiths, D Grossman, Exterior differential systems and Euler–Lagrange partial differential equations, Univ. of Chicago Press (2003) MR1985469
6	Y de Cornulier, Dimension of asymptotic cones of Lie groups, J. Topol. 1 (2008) 342 MR2399134
7	Y de Cornulier, Aspects de la géométrie des groupes, preprint (2020) arXiv:2003.03993
8	Y Cornulier, R Tessera, Contracting automorphisms and L^p-cohomology in degree one, Ark. Mat. 49 (2011) 295 MR2826945
9	T C Dinh, N Sibony, Introduction to the theory of currents, course notes (2005)
10	G Elek, Coarse cohomology and l_p-cohomology, K-Theory 13 (1998) 1 MR1610246
11	A Eskin, D Fisher, K Whyte, Coarse differentiation of quasi-isometries, I : Spaces not quasi-isometric to Cayley graphs, Ann. of Math. 176 (2012) 221 MR2925383
12	L Genton, Scaled Alexander–Spanier cohomology and L^q,p cohomology for metric spaces, PhD thesis, École Polytechnique Fédérale de Lausanne (2014)
13	V Gol’dshtein, M Troyanov, A short proof of the Hölder–Poincaré duality for L_p-cohomology, Rend. Semin. Mat. Univ. Padova 124 (2010) 179 MR2752683
14	M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, II" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 MR1253544
15	J Heber, On the geometric rank of homogeneous spaces of nonpositive curvature, Invent. Math. 112 (1993) 151 MR1207480
16	E Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974) 23 MR353210
17	M Kapovich, B Kleiner, B Leeb, Quasi-isometries and the de Rham decomposition, Topology 37 (1998) 1193 MR1632904
18	A López Neumann, Vanishing of the second L^p-cohomology group for most semisimple groups of rank at least 3, preprint (2023) arXiv:2302.09307
19	P Pansu, Cohomologie L^p : invariance sous quasiisométrie, preprint (1995)
20	P Pansu, Cohomologie L^p, espaces homogènes et pincement, preprint (1999)
21	P Pansu, Cohomologie L^p en degré 1 des espaces homogènes, Potential Anal. 27 (2007) 151 MR2322503
22	P Pansu, Cohomologie L^p et pincement, Comment. Math. Helv. 83 (2008) 327 MR2390047
23	P Pansu, Pincement du plan hyperbolique complexe, preprint (2011) arXiv:1103.4460
24	I Peng, Coarse differentiation and quasi-isometries of a class of solvable Lie groups, I, Geom. Topol. 15 (2011) 1883 MR2860983
25	I Peng, Coarse differentiation and quasi-isometries of a class of solvable Lie groups, II, Geom. Topol. 15 (2011) 1927 MR2860984
26	W Rudin, Functional analysis, McGraw-Hill (1973) MR365062
27	M Rumin, Formes différentielles sur les variétés de contact, J. Differential Geom. 39 (1994) 281 MR1267892
28	R Sauer, M Schrödl, Vanishing of ℓ²-Betti numbers of locally compact groups as an invariant of coarse equivalence, Fund. Math. 243 (2018) 301 MR3863061
29	E Sequeira, Relative L^p-cohomology and application to Heintze groups, Ann. Fenn. Math. 49 (2024) 23 MR4703775
30	L W Tu, An introduction to manifolds, Springer (2008) MR2356311
31	X Xie, Large scale geometry of negatively curved ℝⁿ ⋊ ℝ, Geom. Topol. 18 (2014) 831 MR3180486