L^{2}-torsion of hyperbolic manifolds

Eckehard Hess  Thomas Schick

to appear in Manuscripta Mathematica

Eckehard Hess  Thomas Schick
Fachbereich Mathematik  Fachbereich Mathematik
Universität Mainz  Universität Münster

Einsteinstr. 62

55099 Mainz, Germany  48149 Münster, Germany
hess@topologie.mathematik.uni-mainz.de  thomas.schick@math.uni-muenster.de

The L^{2}-torsion is an invariant defined for compact L^{2}-acyclic manifolds of determinant class, for example odd dimensional hyperbolic manifolds. It was introduced by John Lott [Lot92] and Varghese Mathai [Mat92] and computed for hyperbolic manifolds in low dimensions. Our definition of the L^{2}-torsion coincides with that of John Lott, which is twice the logarithm of that of Varghese Mathai.

In this paper you will find a proof of the fact that the L^{2}-torsion of hyperbolic manifolds of arbitrary odd dimension does not vanish. This was conjectured by John Lott in [Lot92, p.484, Proposition 16 infra]. Some concrete values are computed and an estimate of their growth with the dimension is given. The values we compute for dimensions 5 and 7 differ from those published in [Lot92, Proposition 16]. The result has been independently achieved by both authors and will be part of the dissertation of Eckehard Hess at the university of Mainz. For an introduction into L^{2}-theory see [Lü97].

We are indebted to Prof. Dr. Lück, Münster, for permanent support and encouragement.

1. Definition. Following [Lot92, p.482] we define the analytic L^{2}-torsion of an L^{2}-acyclic Riemannian (d = 2n + 1)-dimensional manifold of determinant class by

Tor_{(2)}(M) = 2 ∑_{j=0}^{n}(−1)^{j+1} logdet_{G}(△_{j}|)

Here G is the fundamental group of M, △_{j}| is the Laplacian restricted to coclosed forms on the universal covering M̃ and the logarithm of the Gdeterminant is computed from the local trace of the heat kernel as follows

logdet_{G}(△_{j}|) = ∫_{F} \{\frac{d}{ds}|_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}tr_{ℂ}e^{−t△_{j}|}(x, x)dt]

+ ∫_{1}^{∞} t^{−1}tr_{ℂ}e^{−t△_{j}|}(x, x)dt}dx

Here F is a fundamental domain of M in M̃, the first integral exists for s sufficiently large and one has to take the meromorphic extension at 0. M being of determinant class ensures the second integral to converge.

2. Theorem. There is a constant α_{d} > 0, such that for every (d = 2n + 1)- dimensional closed hyperbolic manifold

Tor_{(2)}(M) = (−1)^{n}α_{d} Vol(M)

The values for α_{d} have been computed as follows. Although exact values were computed for d ≤ 251, we will give only exact numbers for d ≤ 11 and approximate numbers for d ≤ 39:

\begin{matrix}d & α_{d} & ≈ α_{d} \\ 3 & \frac{1}{3π} & 0, 106103 \\ 5 & \frac{62}{45π^{2}} & 0, 139598 \\ 7 & \frac{221}{35π^{3}} & 0, 203645 \\ 9 & \frac{32204}{945π^{4}} & 0, 349847 \\ 11 & \frac{1339661}{6237π^{5}} & 0, 701891 \end{matrix}  \begin{matrix}d & ≈ α_{d} \\ 13 & 1, 61885 \\ 15 & 4, 22925 \\ 17 & 12, 3578 \\ 19 & 39, 9606 \\ 21 & 141, 729 \\ 23 & 547, 188 \end{matrix}  \begin{matrix}25 & 2284, 87 \\ 27 & 10261, 5 \\ 29 & 49326 \\ 33 & 252701 \\ 35 & 1, 37458 ∗ 10^{6} \\ 37 & 7, 91236 ∗ 10^{6} \\ 39 & 4, 80523 ∗ 10^{7} \end{matrix}

3. Lemma. Let △_{j}| = δ_{j}d_{j}|_{ker(δ_{j+1})} be the Laplacian, restricted to coclosed L^{2}-forms on the (d = 2n + 1)-dimensional hyperbolic space H^{d}. Then for every closed d-dimensional hyperbolic manifold M with fundamental group G and j ≠ n

logdet_{G}(△_{j}|) = Vol(M)C( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}(−1)^{k+1} \frac{2π}{2k+1}(n − j)^{2k+1}

with C = \frac{(4π)^{−(n+ \frac{1}{2} )}}{Γ(n + \frac{1}{2})}  and  a = n − j

Here K_{k,j}^{n} is the coefficient of ν^{2k} in the polynomial

P (ν) := \frac{∏_{i=0}^{n}(ν^{2} + i^{2})}{ν^{2} + (n − j)^{2}} (∗)

Note that P (ν) indeed is a polynomial rather than a rational function, as |a| ∈ {1...n}. In addition

logdet_{G}(△_{n}|) = 0

Proof. Following [Lot92, prop. 15] the local trace of the heat kernel of △_{j}| is

tr_{ℂ}e^{−t△_{j}|}(x, x) = C( \begin{matrix}2n \\ j \end{matrix} ) ∫_{−∞}^{∞} e^{−t(ν^{2}+a^{2})} \frac{∏_{k=0}^{n}(ν^{2} + k^{2})}{ν^{2} + a^{2}}dν

According to the above remark let

\frac{∏_{k=0}^{n}(ν^{2} + k^{2})}{ν^{2} + a^{2}} = ∑_{k=0}^{n} K_{k,j}^{n}ν^{2k}

L^{2}-torsion of hyperbolic manifolds 3 Evaluation of the above integral yields

tr_{ℂ}e^{−t△_{j}}(x, x) = C( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}e^{−ta^{2}}t^{−k− \frac{1}{2}} Γ(k + \frac{1}{2})

Now we have to compute

L_{j} := \frac{logdet(△_{j}|)}{Vol(M)}
= \frac{d}{ds}|_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}tr_{ℂ}e^{−t△_{j}}(x, x)dt] + ∫_{1}^{∞} t^{−1}tr_{ℂ}e^{−t△_{j}}(x, x)dt

John Lott showed in [Lot92, Lemma 13, p.481]

L_{n} = 0

For j ≠ n, that is a = n − j ≠ 0

L_{j} = C( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}Γ(k + \frac{1}{2})\frac{d}{ds}|_{s=0}(\frac{1}{Γ(s)} \underbrace{∫_{0}^{∞} e^{−ta^{2}}t^{s−k− \frac{3}{2}} dt}_{:=J} )

J exists for s sufficiently large. Its meromorphic extension leads to

L_{j} = C( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}Γ(k + \frac{1}{2})Γ( − k − \frac{1}{2})a^{2k+1}
= C( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}(−1)^{k+1} \frac{2π}{2k+1}a^{2k+1}

4. Corollary. For any closed hyperbolic manifold of dimension d = 2n+1 we have by Definition 1

\frac{Tor_{(2)}(M)}{Vol(M)} = 2 ∑_{j=0}^{n−1}(−1)^{j+1}C( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}(−1)^{k+1} \frac{2π}{2k+1}(n − j)^{2k+1}

The numerical values were computed using this fomula and Mathematica.

5. Lemma. Let M be a closed (d = 2n + 1)-dimensional manifold. Then (−1)^{j+1} logdet_{G}(△_{j}|) = (−1)^{n}| logdet_{G}(△_{j}|)|

with

| logdet_{G}(△_{j}|)| > 0 for j ≠ n

In particular

(−1)^{n} Tor_{(2)}(M) > 0

Proof. Let j ≠ n. Then we have

L_{j} = −2πC( \begin{matrix}2n \\ j \end{matrix} ) ∑_{k=0}^{n} K_{k,j}^{n}(−1)^{k} \frac{1}{2k+1}a^{2k+1}

= −2πC( \begin{matrix}2n \\ j \end{matrix} ) ∫_{0}^{a} ∑_{k=0}^{n} K_{k,j}^{n}(ix)^{2k}dx

Using the definition (∗) of the coefficients K_{k,j}^{n} in Lemma 3 one gets

L_{j} = −2πC( \begin{matrix}2n \\ j \end{matrix} ) ∫_{0}^{a} \frac{∏_{k=0}^{n}(k^{2} − x^{2})}{a^{2} − x^{2}}dx

One has

∫_{0}^{a} \frac{∏_{k=0}^{n}(k^{2} − x^{2})}{a^{2} − x^{2}}dx

= (−1)^{n+1} ∫_{0}^{a} \frac{x}{(a + x)(a − x)} ∏_{k=−n}^{n}(x + k)dx

= (−1)^{n+1} ∑_{r=0}^{a−1} ∫_{0}^{1} f_{r}(t)dt

where for t ∈]0, 1[ , r ∈ {0, .., a − 1} we define

f_{r}(t) = \underbrace{\frac{t + r}{(a + t + r)(a − t − r)}}_{>0} ∏_{k=−n+r}^{n+r} \underbrace{(t + k)}_{< 0 for k < 0 ; > 0 otherwise}

= (−1)^{n−r}|f_{r}(t)|

For t ∈ ]0, 1[ and 0 ≤ r < r + 1 ≤ a − 1 one computes

|\frac{f_{r+1}(t)}{f_{r}(t)}| > 1

Hence

∫_{0}^{1} |f_{r+1}(t)|dt ≥ ∫_{0}^{1} |f_{r}(t)|dt

Now the sum

L_{j} = −2πC( \begin{matrix}2n \\ j \end{matrix} ) ∑_{r=0}^{a−1}(−1)^{r+1} ∫_{0}^{1} |f_{r}(t)|dt

is an alternating sum and the absolute values of the summands are stricly increasing. So it is not 0 and the sign is that of the last summand. One concludes

logdet_{G}(△_{j}|) = (−1)^{n−j−1}| logdet_{G}(△_{j}|)|

This also finishes the proof of Theorem 2.

L^{2}-torsion of hyperbolic manifolds 5

6. Proposition. The constants α_{d} of Theorem 2 strictly increase and

α_{2n+1} ≥ \frac{n}{2π} α_{2n−1}

In particular

α_{2n+1} ≥ \frac{2}{3} \frac{n!}{(2π)^{n}}

Proof. An elementary computation shows

|\frac{f_{a−1}(t)}{f_{a−2}(t)}| ≥ 2

Now one has

|∫_{0}^{a} \frac{∏_{k=0}^{n}(k^{2} − x^{2})}{a^{2} − x^{2}}dx| ≥ \frac{1}{2} ∫_{a−1}^{a} |\frac{∏_{k=0}^{n}(k^{2} − x^{2})}{a^{2} − x^{2}} | dx

and

α_{2n+1} ≥ 2π ∑_{j=0}^{n−1} \frac{(4π)^{−(n+ \frac{1}{2} )}}{Γ(n + \frac{1}{2})} ( \begin{matrix}2n \\ j \end{matrix} ) ∫_{n−j−1}^{n−j} |\frac{∏_{k=0}^{n−1}(k^{2} − x^{2})}{(n − j)^{2} − x^{2}} | \underbrace{(n^{2} − x^{2})}_{≥(2n−j)j} dx

≥ 2π ∑_{j=1}^{n−1} \frac{(4π)^{−(n+ \frac{1}{2} )}}{Γ(n + \frac{1}{2})} 2n(2n − 1) ( \begin{matrix}2n − 2 \\ j − 1 \end{matrix} ) ∫_{n−j−1}^{n−j} |\frac{∏_{k=0}^{n−1}(k^{2} − x^{2})}{(n − j)^{2} − x^{2}} | dx

= 4π ∑_{j=1}^{n−1} \frac{2n}{4π} \frac{(4π)^{−(n−1+ \frac{1}{2} )}}{Γ(n − 1 + \frac{1}{2})} ( \begin{matrix}2n − 2 \\ j − 1 \end{matrix} ) ∫_{n−j−1}^{n−j} |\frac{∏_{k=0}^{n−1}(k^{2} − x^{2})}{(n − j)^{2} − x^{2}} | dx

≥ \frac{n}{2π} 4π ∑_{l=0}^{n−2} \frac{(4π)^{−(n−1+ \frac{1}{2} )}}{Γ(n − 1 + \frac{1}{2})} ( \begin{matrix}2(n − 1) \\ l \end{matrix} ) ∫_{n−1−l−1}^{n−1−l} |\frac{∏_{k=0}^{n−1}(k^{2} − x^{2})}{(n − 1 − l)^{2} − x^{2}} | dx

≥ \frac{n}{2π} 4π ∑_{l=0}^{n−2} \frac{(4π)^{−(n−1+ \frac{1}{2} )}}{Γ(n − 1 + \frac{1}{2})} ( \begin{matrix}2(n − 1) \\ l \end{matrix} ) |∫_{0}^{n−1−l} \frac{∏_{k=0}^{n−1}(k^{2} − x^{2})}{(n − 1 − l)^{2} − x^{2}} dx|

= \underbrace{\frac{n}{2π}}_{≥1 for n≥7} α_{2n−1}

For n ≤ 7 the growth follows from the table.

References

[Lü97] W. Lück. L^{2}-invariants of regular coverings of compact manifolds and CW -complexes. In R.B. Daverman, and R.J. Sher, editors, Handbook of Geometry. Elsevier, to appear.

[Lot92] John Lott. Heat kernels on covering spaces and topological invariants. Journal of Differential Geometry, 35:471–510, 1992.

[Mat92] Varghese Mathai. L^{2}-analytic torsion. Journal of Functional Analysis, 107(2):369–386, 1992.