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**The covering spectrum of a compact length space.**
*(English)*
Zbl 1106.58025

J. Differ. Geom. 67, No. 1, 35-77 (2004); Erratum 74, No. 3, 523 (2006).

The authors associate to a complete length space \(X\) a subset of \({\mathbb R}^+\) which is defined by investigating the existence of certain covering spaces of \(X\). Because of its analogy to the length spectrum of a Riemannian manifold, this subset of \({\mathbb R}^+\) is called the covering spectrum of \(X\). Compared to the length spectrum, the covering spectrum has much better behavior under Gromov-Hausdorff limits. There are many interesting isospectrality questions associated to this new kind of spectrum.

Let us give some precise definitions. A complete connected metric space \(X\) is called a complete length space if every pair of points in the space is joined by a length minimizing rectifiable curve. Let \(\delta>0\). We define \(\pi_1(X,\delta,p)\) to be the subgroup of \(\pi_1(X,p)\) generated by the homotopy classes of loops of the form \(\alpha^{-1}\cdot\beta\cdot \alpha\) where \(\beta:[0,1]\to X\) is a rectifiable loop of length \(<\delta\) and where \(\alpha\) is a path from \(p\) to \(\beta(0)=\beta(1)\). Obviously, \(\pi_1(X,\delta,p)\) is a normal subgroup of \(\pi_1(X,p)\). It can be shown that \(X\) has a uniquely determined connected cover \(\pi:\widetilde X^\delta\to X\), \(\pi(\widetilde p)=p\), with \(\pi_*(\pi_1(\widetilde X^\delta,\widetilde p))=\pi_1(X,\delta,p)\). This cover is independent of the choice of \(p\), and is again a complete length space. If \(\rho<\tau\), then \(\widetilde X^\rho\) covers \(\widetilde X^\tau\).

The covering spectrum of \(X\) is the set of all \(\rho>0\) such that the cover \(\widetilde X^\rho\to \widetilde X^\tau\) is a non-trivial cover for all \(\tau>\rho\).

From now on let \(X\) be compact. In this case, it is shown that \(\lim_{\tau\to \rho,\;\tau<\rho} \widetilde X^\tau=\widetilde X^\rho.\) Hence, \(\delta \mapsto \widetilde X^\delta\) is lower semi-continuous, and the covering spectrum contains exactly the positive numbers where this functions fails to be continuous.

The covering spectrum of a compact length space is a discrete subset of the positive real numbers, bounded from above by the diameter of \(X\). It is finite (in other words \(0\) is not an accumulation point), if and only if \(X\) has a universal cover.

The covering spectrum has various relations to the length spectrum and its variants. We want to give some examples and we refer for exact definitions, variants and extension to the article.

For example if \(M\) is a compact length space with a simply connected universal cover and if \(\delta\) is in the covering spectrum, then \(2\delta\) is in the minimum length spectrum of \(X\) which is in turn a subset of the length spectrum. If one replaces the existence of a simply connected universal cover by the weaker assumption that a universal cover \(\widetilde X_U\) exists, then \(2\delta\) is still in the image of the minimum marked length map, which is a “spectral invariant” whose defintion is obtained from the minimum marked length spectrum by using the Deck transformation of \(\widetilde X_U\to X\) instead of \(\pi_1(X)\).

It is also shown that two compact length spaces with universal covers, having the same minimum marked length spectrum, also share the same covering spectrum.

Several sections of the article are devoted to studying the behavior of the covering spectrum under Gromov-Hausdorff limits. The covering spectrum has much better compatibility properties with Gromov-Hausdorff limits than the standard variants of the length spectrum.

One of the main results states: If \(X_i\) is a sequence of compact length spaces converging to a compact length space \(Y\), then the covering spectra of \(X_i\) converge (in a suitable sense of “pointwise” convergence of spectra without multiplicity) to the covering spectrum of \(Y\). As an application it can be deduced that if \(C\) is a Gromov-Hausdorff connected class of compact length spaces with a common discrete length spectrum, then all compact length spaces in the closure of \(C\) have the same covering spectrum as well. As another application one finds uniform gaps in the covering spectrum.

The last section studies the relations of the covering spectrum to the spectrum of the Laplace operator. If \(C\) is a set of Laplace isospectral manifolds which are negatively curved, then there are only finitely many distinct covering spectra for the manifolds in this class. By studying quotients of Heisenberg groups as in [C. S. Gordon, Contemp. Math. 51, 63–80 (1986; Zbl 0591.53042)], one obtains examples of Laplace isospectral manifolds with different covering spectrum. On the other hand, Komatsu pairs of Sunada isospectral manifolds share the same covering spectrum. (Example 10.3 contains an error of minor importance which was corrected by the authors in the Erratum.)

The article is very well written in an essentially self-contained manner. It introduces new and interesting concepts without using heavy machinery. Hence, it is also accessible for young mathematicians who start working on subjects related to length spaces.

Some parts of the article are strongly connected to two other recent articles by the authors [Trans. Am. Math. Soc. 353, No. 9, 3585–3602 (2001; Zbl 1005.53035) and Trans. Am. Math. Soc. 356, No. 3, 1233–1270 (2004; Zbl 1046.53027)]. The article under review can be read independently, but the two additional articles contain important side results.

Let us give some precise definitions. A complete connected metric space \(X\) is called a complete length space if every pair of points in the space is joined by a length minimizing rectifiable curve. Let \(\delta>0\). We define \(\pi_1(X,\delta,p)\) to be the subgroup of \(\pi_1(X,p)\) generated by the homotopy classes of loops of the form \(\alpha^{-1}\cdot\beta\cdot \alpha\) where \(\beta:[0,1]\to X\) is a rectifiable loop of length \(<\delta\) and where \(\alpha\) is a path from \(p\) to \(\beta(0)=\beta(1)\). Obviously, \(\pi_1(X,\delta,p)\) is a normal subgroup of \(\pi_1(X,p)\). It can be shown that \(X\) has a uniquely determined connected cover \(\pi:\widetilde X^\delta\to X\), \(\pi(\widetilde p)=p\), with \(\pi_*(\pi_1(\widetilde X^\delta,\widetilde p))=\pi_1(X,\delta,p)\). This cover is independent of the choice of \(p\), and is again a complete length space. If \(\rho<\tau\), then \(\widetilde X^\rho\) covers \(\widetilde X^\tau\).

The covering spectrum of \(X\) is the set of all \(\rho>0\) such that the cover \(\widetilde X^\rho\to \widetilde X^\tau\) is a non-trivial cover for all \(\tau>\rho\).

From now on let \(X\) be compact. In this case, it is shown that \(\lim_{\tau\to \rho,\;\tau<\rho} \widetilde X^\tau=\widetilde X^\rho.\) Hence, \(\delta \mapsto \widetilde X^\delta\) is lower semi-continuous, and the covering spectrum contains exactly the positive numbers where this functions fails to be continuous.

The covering spectrum of a compact length space is a discrete subset of the positive real numbers, bounded from above by the diameter of \(X\). It is finite (in other words \(0\) is not an accumulation point), if and only if \(X\) has a universal cover.

The covering spectrum has various relations to the length spectrum and its variants. We want to give some examples and we refer for exact definitions, variants and extension to the article.

For example if \(M\) is a compact length space with a simply connected universal cover and if \(\delta\) is in the covering spectrum, then \(2\delta\) is in the minimum length spectrum of \(X\) which is in turn a subset of the length spectrum. If one replaces the existence of a simply connected universal cover by the weaker assumption that a universal cover \(\widetilde X_U\) exists, then \(2\delta\) is still in the image of the minimum marked length map, which is a “spectral invariant” whose defintion is obtained from the minimum marked length spectrum by using the Deck transformation of \(\widetilde X_U\to X\) instead of \(\pi_1(X)\).

It is also shown that two compact length spaces with universal covers, having the same minimum marked length spectrum, also share the same covering spectrum.

Several sections of the article are devoted to studying the behavior of the covering spectrum under Gromov-Hausdorff limits. The covering spectrum has much better compatibility properties with Gromov-Hausdorff limits than the standard variants of the length spectrum.

One of the main results states: If \(X_i\) is a sequence of compact length spaces converging to a compact length space \(Y\), then the covering spectra of \(X_i\) converge (in a suitable sense of “pointwise” convergence of spectra without multiplicity) to the covering spectrum of \(Y\). As an application it can be deduced that if \(C\) is a Gromov-Hausdorff connected class of compact length spaces with a common discrete length spectrum, then all compact length spaces in the closure of \(C\) have the same covering spectrum as well. As another application one finds uniform gaps in the covering spectrum.

The last section studies the relations of the covering spectrum to the spectrum of the Laplace operator. If \(C\) is a set of Laplace isospectral manifolds which are negatively curved, then there are only finitely many distinct covering spectra for the manifolds in this class. By studying quotients of Heisenberg groups as in [C. S. Gordon, Contemp. Math. 51, 63–80 (1986; Zbl 0591.53042)], one obtains examples of Laplace isospectral manifolds with different covering spectrum. On the other hand, Komatsu pairs of Sunada isospectral manifolds share the same covering spectrum. (Example 10.3 contains an error of minor importance which was corrected by the authors in the Erratum.)

The article is very well written in an essentially self-contained manner. It introduces new and interesting concepts without using heavy machinery. Hence, it is also accessible for young mathematicians who start working on subjects related to length spaces.

Some parts of the article are strongly connected to two other recent articles by the authors [Trans. Am. Math. Soc. 353, No. 9, 3585–3602 (2001; Zbl 1005.53035) and Trans. Am. Math. Soc. 356, No. 3, 1233–1270 (2004; Zbl 1046.53027)]. The article under review can be read independently, but the two additional articles contain important side results.

Reviewer: Bernd Ammann (Vandœuvre-les-Nancy)