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Sol Manifolds

SOL is one of the eight Thurston's geometries of three dimensional manifolds. Topologically it is R3\mathbb{R}^3 with the metric

ds2=e−2zdx2+e2zdy2+dz2.ds^2 = e^{-2z}dx^2 + e^{2z}dy^2 + dz^2.

It also is a solvable Lie group, such that the exponential map sol→SOL\mathfrak{sol} \to \mathrm{SOL} is a diffeomorphism and the metric above described is left invariant with respect to the group operation

(x,y,z)⋅(x′,y′,z′)=(x+ezx′,y+e−zy′,t+t′).(x,y,z) \cdot (x',y',z') = (x+ e^z x',y+ e^{-z} y', t + t').

Volume entropy

We aim to compute the volume entropy of the group, defined as

δSOL=lim⁡r→∞1rlog⁡Vol⁡(B(0,r)),\delta_{\mathrm{SOL}} = \lim_{r \to \infty} \frac{1}{r} \log \operatorname{Vol}(B(0,r)),

where B(0,r)B(0,r) is the geodesic ball centred at the origin of radius rr. For the compuation, we follow Schwartz and notice the projection ηX:SOL→R2\eta_X: \mathrm{SOL} \to \mathbb{R}^2, (x,y,z)↦(0,y,z)(x, y, z) \mapsto (0, y, z) decreases areas and sends the geodesic sphere Sr=∂B(0,r)\mathcal{S}_r = \partial B(0, r) onto the hyperbolic disk D(0,r)={(y,z)∈R2∣d(0,(0,y,z))<r}\mathbb{D}(0, r) = \{(y, z) \in \mathbb{R}^2 \mid d(0, (0, y, z)) < r\}. The area of the hyperbolic disk is 4πsinh⁡(r/2)24\pi\sinh(r/2)^2 and the projection is two-to-one, whence

8πsinh⁡(r/2)2≤Area⁡(Sr).8\pi\sinh(r/2)^2 \leq \operatorname{Area}(\mathcal{S}_r).

On the other hand,

8πsinh⁡(r/2)2=2π(cosh⁡(r)−1),8\pi\sinh(r/2)^2 = 2\pi(\cosh(r) - 1),

and the volume of the geodesic ball is

Vol⁡(Br)=∫0rArea⁡(St)dt,\operatorname{Vol}(B_r) = \int_0^r \operatorname{Area}(\mathcal{S}_t) dt,

we conclude,

2π(sinh⁡(r)−r)≤Vol⁡(B(0,r)).2\pi (\sinh(r) - r) \leq \operatorname{Vol}(B(0, r)).

From this inequality, is follows that 1≤δSOL1 \leq \delta_{\mathrm{SOL}}.

An upper bound for the entropy

The upper bound for the volume entropy is a work of Schwartz and Kopczynski, their argument is remarkably simple, it comes from the observation that every point (x,y,z)∈Sr(x, y, z) \in \mathcal{S}_r satisfies the inequalitites,

∣x∣,∣y∣≤er+r,∣z∣≤r,(∣x∣−r)(∣y∣−r)≤er,\begin{align} |x|, |y| \leq e^r + r, && |z| \leq r, && (|x| - r)(|y| - r) \leq e^r, \end{align}

then they estimate the upper bound 72r2er72r^2e^r for Vol⁡(B(0,r))\operatorname{Vol}(B(0,r)), by this upper bound, together with the lower bound of the previous section, they conclude that the volume entropy of SOL⁡\operatorname{SOL} is 1.

References