Roe $C^*$-algebra for groupoids and generalized Lichnerowicz vanishing theorem for foliated manifolds

Abstract

We introduce the concept of Roe $C^*$-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes’ tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe $C^*$-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.

Appendix

In this Appendix, we prove the following lemma. Such a result is known to experts, but we could not find a suitable reference so include a proof here for the reader’s convenience.

Lemma 6.1

Let $\iota$, $\kappa$, r, and R be positive constants, and d a positive integer. Then for any $R>0$, there exists $m\in \mathbb {N}$ depending only on $\iota , \kappa , r, R$, and d with the following property. Let M be a d-dimensional complete Riemannian manifold with all sectional curvatures in the interval $[-\kappa , \kappa ]$ and injectivity radius bounded below by $\iota$. There exists a cover $\mathcal {U}$ of M by balls of radius r such that for any $U\in \mathcal {U}$ there are at most m balls $V\in \mathcal {U}$ with

$$\begin{aligned} V\cap N_R(U)\ne \varnothing , \end{aligned}$$

where $N_R(U):=\{y\in M | d(y, U)<R\}$. Moreover, if $r<\iota$, then each $U\in \mathcal {U}$ is diffeomorphic to ${\mathbb {R}}^d$.

Proof

Using Zorn’s lemma, we can choose a maximal subset Z of M such that $d(x,y)\ge r$ for all $x,y\in Z$. Let

$$\begin{aligned} {\mathcal U}:=\{B(z; r) \mid z\in Z\}, \end{aligned}$$

where B(z; r) is the geodesic ball center at z with radius r. If $r<\iota$, each B(z; r) is diffeomorphic to an open ball in $\mathbb {R}^d$ via the exponential map. Note that as Z is maximal, the union of B(z; r) in ${\mathcal U}$ must cover the whole M, and so ${\mathcal U}$ is an open cover of M.

Fix $U=B(z; r)\in {\mathcal U}$, and let $S_z$ be the set of $y\in Z$ such that

$$\begin{aligned} B(y;r)\cap B(z; R+r)\ne \varnothing . \end{aligned}$$

We consider the collection of B(y; r / 2) for $y\in S_z$. The triangle inequality shows that for all $y\in S_z$,

$$\begin{aligned} B(z; R+3r)\supseteq B(y; r)\supseteq B(y; r/2). \end{aligned}$$

Furthermore, as the points in $S_z$ are at least r-distance apart, the sets $\{B(y;r/2)\mid y\in S_z\}$ are mutually disjoint. Hence we have the following inclusion

$$\begin{aligned} B(z; R+3r)\supseteq \bigsqcup _{z_i\in S_z} B(z_i; r/2). \end{aligned}$$

By the Bishop-Gromov theorem [4, Theorem 107, P. 310], there is a constant $C>0$ depending only only $\kappa$, d, and $R+3r$ such that

$$\begin{aligned} {\text {Volume}}(B(z; R+3r))\le C. \end{aligned}$$

On the other hand, a standard result in comparison theory [4, Theorem 103, P. 306] gives us a constant $c>0$ depending only on r, $\kappa$, $\iota$ and d such that

$$\begin{aligned} {\text {Volume}}(B(y; r/2))\ge {\text {Volume}}(B(y;\min (r/2,\iota ))\ge c, \end{aligned}$$

for each $y\in S_z$.

Combing these two volume bounds, we see that

$$\begin{aligned} C\ge {\text {Volume}}(B(z; R+3r)) \ge {\text {Volume}}\left( \bigsqcup _{y\in S_z}B(y; r/2)\right) \ge |S_z| c. \end{aligned}$$

We conclude that $|S_z|\le C/c$, and note that C / c only depends on $\iota$, $\kappa$, R, r, and d. We choose m to be an integer greater than or qual to C / c, and therefore complete the proof.