The Cheeger constant of simply connected, solvable Lie groups

Peyerimhoff, Norbert; Samiou, Evangelia

doi:10.1090/S0002-9939-03-07404-5

The Cheeger constant of simply connected, solvable Lie groups
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by Norbert Peyerimhoff and Evangelia Samiou PDF

Proc. Amer. Math. Soc. 132 (2004), 1525-1529 Request permission

Abstract:

We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group $G$ with Lie algebra $\mathfrak {G}$ is \[ h(G)=\max _{H\in \mathfrak {G},||H||=1} \textrm {tr }(\textrm {ad }(H)).\]

References

Robert Brooks, The spectral geometry of a tower of coverings, J. Differential Geom. 23 (1986), no. 1, 97–107. MR 840402
Robert Brooks, Peter Perry, and Peter Petersen V, On Cheeger’s inequality, Comment. Math. Helv. 68 (1993), no. 4, 599–621. MR 1241474, DOI 10.1007/BF02565837
Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
Christopher Connell, Asymptotic harmonicity of negatively curved homogeneous spaces and their measures at infinity, Comm. Anal. Geom. 8 (2000), no. 3, 575–633. MR 1775140, DOI 10.4310/CAG.2000.v8.n3.a7
Alexander Grigor′yan, Estimates of heat kernels on Riemannian manifolds, Spectral theory and geometry (Edinburgh, 1998) London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 140–225. MR 1736868, DOI 10.1017/CBO9780511566165.008
Harry F. Hoke III, Lie groups that are closed at infinity, Trans. Amer. Math. Soc. 313 (1989), no. 2, 721–735. MR 935533, DOI 10.1090/S0002-9947-1989-0935533-2
G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. 7 (1997), no. 4, 755–782. MR 1465601, DOI 10.1007/s000390050025
Enrico Leuzinger, Kazhdan’s property (T), $L^2$-spectrum and isoperimetric inequalities for locally symmetric spaces, Comment. Math. Helv. 78 (2003), no. 1, 116–133. MR 1966754, DOI 10.1007/s000140300005
Anthony Manning, Topological entropy for geodesic flows, Ann. of Math. (2) 110 (1979), no. 3, 567–573. MR 554385, DOI 10.2307/1971239
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
Norbert Peyerimhoff, Isoperimetric and ergodic properties of horospheres in symmetric spaces, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 797–808. MR 1858556, DOI 10.1090/pspum/069/1858556
Ch. Pittet, The isoperimetric profile of homogeneous Riemannian manifolds, J. Differential Geom. 54 (2000), no. 2, 255–302. MR 1818180
R. J. Spatzier, Dynamical properties of algebraic systems, Dissertation, Warwick and Maryland, 1983.