Problem Sets

Recall that the disc model of \(\mathbb{H}\) is the open unit disc \(D=\{\|v\|<1\}\) equipped with the Riemannian metric \[dg^2 = \frac{4}{(1-\|v\|^2)^2} dv^2\] and the upper half-plane model is the plane \(U=\{v=(x,y)\in \R^2\mid y>0\}\) with metric \[dg^2 = \frac{dv^2}{y^2}.\]

1. Show that for any \(\theta<\frac{\pi}{3}\), there is an equilateral triangle in \(\mathbb{H}\) with all angles equal to \(\theta\). (When \(\theta=\frac{2\pi}{n}\), these triangles tile \(\mathbb{H}\).)

   Show that there is a hyperbolic octagon with all angles equal to \(\frac{\pi}{4}\) and use it to construct a closed surface with sectional curvature \(-1\).

2. A turtle on the hyperbolic plane walks \(x_1\) units forward, turns 90 degrees, walks \(x_2\) units forward, turns 90 degrees, walks \(x_3\) units forward, turns 90 degrees, and finally walks \(x_4\) units forward. Estimate how far the turtle is from its starting point.
3. Show that if we view \(\R^2\) as the complex plane, then for any \(a,b,c,d\in \R\) such that \(ad-bc=1\), the map \(f(z)=\frac{az+b}{cz+d}\) is an isometry of the upper half-plane model.
4. Put a metric on the infinite strip \([0,\pi]\times \R\) that makes it isometric to the hyperbolic plane by a conformal map (a map that preserves angles). Some complex analysis may help here.
5. Show that any \(n\)–gon in \(\mathbb{H}\) has area at most \((n-2)\pi\).
6. Show that there is a \(C>0\) such that any simple closed curve of length \(L\) in \(\mathbb{H}\) bounds a disc of area at most \(CL\).

1. • Show that the logarithmic spiral \(\alpha\from (0,\infty)\to \R^2\), \[\alpha(t) = (t\cos \log t, t \sin \log t)\] is a quasi-geodesic.
   • Construct a quasi-isometry from \(\R^2\) to itself that sends a geodesic ray from the origin to \(\alpha\). Conclude that quasi-isometries of \(\R^2\) need not induce a map on the ideal boundary.

2. (Project) Let \(S\) be the surface formed by gluing a hyperbolic octagon as in problem 1 of the first problem set. The universal cover of \(S\) is the hyperbolic plane, and the covering map gives a tiling of the hyperbolic plane by copies of the hyperbolic octagon. Use a computer to draw this tiling.

   If you cut \(S\) in half along a closed geodesic and color one half red and the other half blue, this corresponds to coloring each of the octagons half red and half blue. What does the resulting coloring of the hyperbolic plane look like? Show that each single-colored component is convex in the hyperbolic metric.

3. Let \(G\) be a finitely generated group with a finite generating set \(S\) and let \(\gamma\) be its growth function with respect to \(S\).
   • Show that for all \(a,b\ge 0\), \(\gamma(a+b)\le \gamma(a)\gamma(b)\).
   • Show that the growth exponent \(e_S=\lim_{n\to \infty}\frac{\log \gamma(n)}{n}\) exists.
4. Prove some of the properties of ultralimits stated in class, for instance:
   • the ultralimit of a sequence is unique
   • \(\lim_\omega x_i+y_i = \lim_\omega x_i + \lim_\omega y_i\)
   • If \(x_i\ge y_i\) for all but finitely many \(i\), then \(\lim_\omega x_i\ge \lim_\omega y_i\)