# zbMATH — the first resource for mathematics

Harmonic analysis, cohomology, and the large-scale geometry of amenable groups. (English) Zbl 1064.43004
This paper explores aspects of geometric group theory involving representation theory and cohomology. Two groups $$\Gamma$$ and $$\Lambda$$ generated by finite symmetric sets are called quasi-isomorphic if there is a map $$\phi$$ from $$\Lambda$$ to $$\Gamma$$ and constants $$\alpha\geq 1$$ and $$K\geq 0$$ such that for all $$\lambda_1$$, $$\lambda_2$$ in $$\Lambda$$ $\alpha^{-1} d_\Lambda(\lambda_1, \lambda_2)- K\leq d_\Gamma(\phi\lambda_1, \phi\lambda_2)\leq \alpha d_\Lambda(\lambda_1, \lambda_2)+ K$ and any element of $$\Lambda$$ is within distance $$\leq K$$ from $$\phi\Lambda$$. Here $$d_\Lambda$$ and $$d_\Gamma$$ are the word metrics on the respective groups. The author explores various implications of quasi-isometry to a host of settings.
The first result is that if $$\Gamma$$ is quasi-isomorphic to $$\mathbb{Z}^d$$, then $$\Gamma$$ has a finite index subgroup isomorphic to $$\mathbb{Z}^d$$. Other results that bring in cohomology for quasi-isomorphic groups are as follows: If $$\Gamma$$ and $$\Lambda$$ are finitely generated, then all of their respective Betti numbers are equal. If $$\Gamma$$ is quasi-isomorphic to a polycyclic group, then all of its virtual Betti numbers are greater than zero. The cohomological dimensions of $$\Gamma$$ and $$\Lambda$$ over a ring $$R$$ are identical. The author also applies his techniques to a class of non-finitely presentable abelian-by-cyclic groups.
The author discusses a notion more general than quasi-isometry, namely, uniform embedding. Many of the results for quasi-isomorphic groups hold for uniformly embedded pairs of groups. For example, cohomological dimensions are preserved. Having the type (FP) property and for solvable groups $$\Gamma$$ and $$\Lambda$$, the Hirsh number of $$\Lambda$$ is bounded by that of $$\Gamma$$.
Let $$\Gamma$$ be a discrete group. Definition: $$\Gamma$$ has property $$H_{FD}$$ if for every unitary representation $$\pi$$ of $$\Gamma$$ with non-zero first cohomology with respect to $$\pi$$ there exists a finite-dimensional $$\Gamma$$-subrepresentation. The author proves that among amenable groups, the property $$H_{FD}$$ is a quasi-isometry invariant.

##### MSC:
 43A07 Means on groups, semigroups, etc.; amenable groups 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 20J06 Cohomology of groups
Full Text:
##### References:
 [1] Assouad, P., Plongements lipschitziens dansR n .Bull. Soc. Math. France, 111 (1983), 429–448. · Zbl 0597.54015 [2] Baumslag, G., Wreath products and finitely presented groups.Math. Z., 75 (1960/1961). 22–28. · Zbl 0090.24402 [3] Benjamini, I. &Schramm, O., Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant.Geom. Funct. Anal., 7 (1997), 403–419. · Zbl 0882.05052 [4] Benoist, Y., Private communication (available upon request). [5] Bergelson, V., Weakly mixing PET.Ergodic Theory Dynam Systems, 7 (1987), 337–349. · Zbl 0645.28012 [6] Bergelson, V. &Rosenblatt, J., Mixing actions of groups.Illinois J. Math., 32 (1988), 65–80. · Zbl 0619.43005 [7] Bestvina, M. &Feighin, M., Proper actions of lattices on contractible manifolds.Invent. Math., 150 (2002), 237–256. · Zbl 1041.57015 [8] Bestvina, M., Kapovich, M. &Kleiner, B., van Kampen’s embedding obstruction for discrete groups.Invent. Math., 150 (2002), 219–235. · Zbl 1041.57016 [9] Bieri, R.,Homological Dimensibn of Discrete Groups. Queen Mary College Mathematics Notes, Queen Mary College, London, 1976. · Zbl 0357.20027 [10] Bieri, R. &Strebel, R., Almost finitely presented soluble groups.Comment. Math. Helv., 53 (1978), 258–278. · Zbl 0373.20035 [11] Block, J. &Weinberger, S., Large scale homology theories and geometry, inGeometric Topology (Athens, GA, 1993), pp. 522–569. AMS/IP Stud. Adv. Math., 2.1., Amer. Math. Soc., Providence, RI, 1997. · Zbl 0898.55006 [12] Borel, A. &Serre, J.-P., Corners and arithmetic groups.Comment. Math. Helv., 48 (1973), 436–491. · Zbl 0274.22011 [13] Bourdon, M. &Pajot, H., Rigidity of quasi-isometries for some hyperbolic buildings.Comment. Math. Helv., 75 (2000), 701–736. · Zbl 0976.30011 [14] Bridson, M. R. &Gersten, S. M., The optimal isoperimetric inequality for torus bundles over the circle.Q. J. Math., 47 (1996), 1–23. · Zbl 0852.20031 [15] Chou, C., Elementary amenable groups.Illinois J. Math., 24 (1980), 396–407. · Zbl 0439.20017 [16] Delorme, P., 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations.Bull. Soc. Math. France, 105 (1977), 281–336. · Zbl 0404.22006 [17] Drutu, C., Quasi-isometry invariants and asymptotic cones.Internat. J. Algebra Comput., 12 (2002), 99–135. · Zbl 1010.20029 [18] Dunwoody, M. J., Accessibility and groups of cohomological dimension one.Proc. London Math. Soc., 38 (1979), 193–215. · Zbl 0419.20040 [19] Dyubina, A., Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups.Int. Math. Res. Not., 2000 (2000), 1997–1101. · Zbl 0979.20034 [20] Erschler, A., On isoperimetric profiles of finitely generated groups.Geom. Dedicata, 100 (2003), 151–171. · Zbl 1049.20024 [21] Farb, B., The quasi-isometry classification of lattices in semisimple Lie groups.Math. Res. Lett., 4 (1997), 705–717. · Zbl 0889.22010 [22] Farb, B. &Mosher, L., A rigidity theorem for the solvable Baumslag-Solitar groups.Invent. Math., 131 (1998), 419–451. · Zbl 0937.22003 [23] –, Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II.Invent. Math., 137 (1999), 613–649. · Zbl 0931.20035 [24] –, On the asymptotic geometry of abelian-by-cyclic groups.Acta Math., 184 (2000), 145–202. · Zbl 0982.20026 [25] –, Problems on the geometry of finitely generated solvable groups, inCrystallographic Groups and their Generalizations (Kortrijk, 1999), pp. 121–134. Contemp. Math., 262. Amer. Math. Soc., Providence, RI, 2000. [26] –, The geometry of surface-by-free groups.Geom. Funct. Anal., 12 (2002), 915–963. · Zbl 1048.20026 [27] Gersten, S. M., Quasi-isometry invariance of cohomological dimension.C. R. Acad. Sci. Paris Sér. I. Math., 316 (1993), 411–416. · Zbl 0805.20043 [28] Ghys, É. &de la Harpe, P. (editors),Sur les groupes hyperboliques d’après Mikhael Gromov. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. Progr. Math., 83. Birkhäuser, Boston, MA, 1990. [29] Gildenhuys, D. &Strebel, R., On the cohomology of soluble groups, II.J. Pure Appl. Algebra, 26 (1982), 293–323. · Zbl 0493.20032 [30] Gromov, M., Groups of polynomial growth and expanding maps.Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73. · Zbl 0474.20018 [31] –, Asymptotic invariants of infinite groups, inGeometric Group Theory, Vol. 2 (Sussex, 1991), pp. 1–295. London Math. Soc. Lecture Note Ser., 182. Cambridge Univ. Press, Cambridge, 1993. [32] Guichardet, A.,Cohomologie des groupes topologiques et des algèbres de Lie. Textes Mathématiques, 2. CEDIC, Paris, 1980. · Zbl 0464.22001 [33] de la Harpe, P.,Topics in Geometric Group Theory. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. · Zbl 0965.20025 [34] de la Harpe, P. &Valette, A.,La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque, 175. Soc. Math. France, Paris, 1989. [35] Kapovich, M. & Kleiner, B., Coarse Alexander duality and duality groups. To appear inJ. Differential Geom. · Zbl 1086.57019 [36] Korevaar, N. J. &Schoen, R. M., Global existence theorems for harmonic maps to non-locally compact spaces.Comm. Anal. Geom., 5 (1997), 333–387. · Zbl 0908.58007 [37] Kropholler, P. H., Cohomological dimension of soluble groups.J. Pure Appl. Algebra, 43 (1986), 281–287. · Zbl 0603.20033 [38] Macdonald, I. D.,The Theory of Groups. Reprint of the 1968 original. Krieger, Malabar, FL, 1988. [39] Merzlyakov, Yu. I., Locally solvable groups of finite rank.Algebra i Logika Sem., 3 (1964), 5–16 (Russian). [40] Milnor, J., Growth of finitely generated solvable groups.J. Differential Geom., 2 (1968), 447–449. · Zbl 0176.29803 [41] Monod, N. & Shalom, Y., Orbit equivalence rigidity and bounded cohomology. To appear inAnn. of Math. · Zbl 1129.37003 [42] Montgomery, D. &Zippin, L.,Topological Transformation Groups. Reprint of the 1955 original. Krieger, Huntington, NY, 1974. · Zbl 0323.57023 [43] Mosher, L., Sageev, M. &Whyte, K., Quasi-actions on trees, I: Bounded valence.Ann. of Math., 158 (2003), 115–164. · Zbl 1038.20016 [44] Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.Ann. of Math., 129 (1989), 1–60. · Zbl 0678.53042 [45] Raghunathan, M. S.,Discrete Subgroups of Lie Groups. Ergeb. Math. Grenzgeb., 68. Springer-Verlag, New York-Heidelberg, 1972. · Zbl 0254.22005 [46] Reiter Ahlin, A., The large scale geometry of nilpotent-by-cyclic groups. Ph.D. Thesis, University of Chicago, 2002. · Zbl 1009.20033 [47] Rosenblatt, J. M., Invariant measures and growth conditions.Trans. Amer. Math. Soc., 193 (1974), 33–53. · Zbl 0246.28017 [48] Sela, Z., Uniform embeddings of hyperbolic groups in Hilbert spaces.Israel J. Math., 80 (1992), 171–181. · Zbl 0785.46032 [49] Shalom, Y., The growth of linear groups.J. Algebra, 199 (1998), 169–174. · Zbl 0892.20023 [50] –, Rigldity of commensurators and irreducible lattices.Invent. Math., 141 (2000), 1–54. · Zbl 0978.22010 [51] Stammbach, U., On the weak homological dimension of the group algebra of solvable groups.J. London Math. Soc., 2 (1970), 567–570. · Zbl 0204.35302 [52] Tits, J., Appendix to ”Groups of polynomial growth and expanding maps” by M. Gromov.Inst. Hautes Études Sci. Publ. Math., 53 (1981), 74–78. [53] Vershik, A. M. &Karpushev, S. I., Cohomology of groups in unitary representations, neighborhood of the identity and conditionally positive definite functions.Mat. Sb., 119 (161), (1982), 521–533. (Russian); English translation inMath. USSR-Sb., 47 (1984), 513–526. · Zbl 0513.43009 [54] Wolf, J. A., Growth of finitely generated solvable groups and curvature of Riemannian manifolds.J. Differential Geom., 2 (1968), 421–446. · Zbl 0207.51803 [55] Zimmer, R. J.,Ergodic Theory and Semisimple Groups. Monographs Math., 81. Birkhäuser, Basel, 1984. · Zbl 0571.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.