On polyhedral retracts and compactifications of locally symmetric spaces. (English) Zbl 1052.22008

Leuzinger, Enrico

Differ. Geom. Appl. 20, No. 3, 293-318 (2004).

Let \(V:=\Gamma\setminus X\) be a locally symmetric space, where \(X\) is a Riemannian symmetric space of noncompact type and rank at least two and \(\Gamma\) a nonuniform, torsion-free, irreducible lattice in the group of isometries of \(X\). The results in this paper are based on the author’s previous construction of the exhausting of \(V\) by Riemannian polyhedra \(V(s)\) (the Riemannian metric is the one induced from \(V\)), \(s\geq 0\), i.e., compact submanifolds with corners [Invent. Math. 121, 389–410 (1995; Zbl 0844.53040)]. The author proves that each \(V(s)\) is a strong deformation retract of \(V\) and \(V\) is homeomorphic to the interior of any \(V(s)\), which thus yields a compactification of \(V\). Equivalent such retractions were independently constructed by L. Saper [Comment. Math. Helv. 72, 167–202 (1997; Zbl 0890.22003)].
This compactification approach is in contrast to other compactifications introduced by various authors [W. Baily and A. Borel, Ann. Math. (2) 84, 442–528 (1966; Zbl 0154.08602); A. Borel and J.-P. Serre, Comment. Math. Helv. 48, 436–491 (1973; Zbl 0274.22011); L. Saper, loc. cit.; I. Satake, Ann. Math. (2) 72, 555–580 (1960; Zbl 0146.04701); C. L. Siegel, Zur Reduktionstheorie quadratischer Formen, Publ. Math. Soc. Japan 5 (1959; Zbl 0097.00901)].
Furthermore one shows that each retraction \(r_s:V\to V(s)\) has a \(\Gamma\)-invariant lift \(\hat{r}_s:X\to X(s)\), where \(X(s)\) is the universal covering space of \(V(s)\), and consequently the lattice \(\Gamma\) is isomorphic to \(\pi_1(V(s))\). The submanifold with corners \(X(s)\) of \(X\) is quasi-isometric to the discrete group \(\Gamma\) equipped with a left-invariant word metric. As an application the proof of a result announced by M. Gromov [Geometric group theory (1993; Zbl 0841.20039)] asserting that the filling-volume functions of irreducible lattices in semisimple Lie groups of \(\mathbb{R}\)-rank at least two have exponential upper bounds in all dimensions is given. The relation of the above compactification of \(V\) with the Borel-Serre compactification of \(V\) is discussed.

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

22E40	Discrete subgroups of Lie groups
53C35	Differential geometry of symmetric spaces
53C20	Global Riemannian geometry, including pinching

Keywords:

Riemannian symmetric space; locally symmetric space; discrete group; semisimple Lie group; arithmetic group; reduction theory; Busemann function; polyhedral retract; Borel-Serre compactification; horoball; filling-volume function; Gromov-Hausdorff limit

Citations:

Zbl 0844.53040; Zbl 0890.22003; Zbl 0154.08602; Zbl 0274.22011; Zbl 0146.04701; Zbl 0097.00901; Zbl 0841.20039

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