Geometry Of Group Actions

# Geometry Of Group Actions

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Paper 1, Section I, F

commentExplain what it means to say that $G$ is a crystallographic group of isometries of the Euclidean plane and that $\bar{G}$ is its point group. Prove the crystallographic restriction: a rotation in such a point group $\bar{G}$ must have order $1,2,3,4$ or 6 .

Paper 1, Section II, F

commentFor which circles $\Gamma$ does inversion in $\Gamma$ interchange 0 and $\infty$ ?

Let $\Gamma$ be a circle that lies entirely within the unit $\operatorname{disc} \mathbb{D}=\{z \in \mathbb{C}:|z|<1\} .$ Let $K$ be inversion in this circle $\Gamma$, let $J$ be inversion in the unit circle, and let $T$ be the Möbius transformation $K \circ J$. Show that, if $z_{0}$ is a fixed point of $T$, then

$J\left(z_{0}\right)=K\left(z_{0}\right)$

and this point is another fixed point of $T$.

By applying a suitable isometry of the hyperbolic plane $\mathbb{D}$, or otherwise, show that $\Gamma$ is the set of points at a fixed hyperbolic distance from some point of $\mathbb{D}$.

Paper 2, Section I, F

commentShow that a map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is an isometry for the Euclidean metric on the plane $\mathbb{R}^{2}$ if and only if there is a vector $\boldsymbol{v} \in \mathbb{R}^{2}$ and an orthogonal linear map $B \in \mathrm{O}(2)$ with

$T(\boldsymbol{x})=B(\boldsymbol{x})+\boldsymbol{v} \quad \text { for all } \boldsymbol{x} \in \mathbb{R}^{2}$

When $T$ is an isometry with $\operatorname{det} B=-1$, show that $T$ is either a reflection or a glide reflection.

Paper 3, Section I, F

commentLet $U$ be a "triangular" region in the unit disc $\mathbb{D}$ bounded by three hyperbolic geodesics $\gamma_{1}, \gamma_{2}, \gamma_{3}$ that do not meet in $\mathbb{D}$ nor on its boundary. Let $J_{k}$ be inversion in $\gamma_{k}$ and set

$A=J_{2} \circ J_{1} ; \quad B=J_{3} \circ J_{2} .$

Let $G$ be the group generated by the Möbius transformations $A$ and $B$. Describe briefly a fundamental set for the group $G$ acting on $\mathbb{D}$.

Prove that $G$ is a free group on the two generators $A$ and $B$. Describe the quotient surface $\mathbb{D} / G$.

Paper 4, Section I, F

commentDefine loxodromic transformations and explain how to determine when a Möbius transformation

$T: z \mapsto \frac{a z+b}{c z+d} \quad \text { with } \quad a d-b c=1$

is loxodromic.

Show that any Möbius transformation that maps a disc $\Delta$ onto itself cannot be loxodromic.

Paper 4, Section II, F

commentExplain briefly how Möbius transformations of the Riemann sphere are extended to give isometries of the unit ball $B^{3} \subset \mathbb{R}^{3}$ for the hyperbolic metric.

Which Möbius transformations have extensions that fix the origin in $B^{3}$ ?

For which Möbius transformations $T$ can we find a hyperbolic line in $B^{3}$ that $T$ maps onto itself? For which of these Möbius transformations is there only one such hyperbolic line?

Paper 1, Section I, F

commentExplain what is meant by stereographic projection from the 2-dimensional sphere to the complex plane.

Prove that $u$ and $v$ are the images under stereographic projection of antipodal points on the sphere if and only if $u \bar{v}=-1$.

Paper 1, Section II, F

commentDefine frieze group and crystallographic group and give three examples of each, identifying them as abstract groups as well as geometrically.

Let $G$ be a discrete group of isometries of the Euclidean plane which contains a translation. Prove that $G$ contains no element of order 5 .

Paper 2, Section $\mathbf{I}$, F

commentDescribe the geodesics in the hyperbolic plane (in a model of your choice).

Let $l_{1}$ and $l_{2}$ be geodesics in the hyperbolic plane which do not meet either in the plane or at infinity. By considering the action on a suitable third geodesic, or otherwise, prove that the composite $R_{l_{1}} \circ R_{l_{2}}$ of the reflections in the two geodesics has infinite order.

Paper 3, Section I, F

commentExplain why there are discrete subgroups of the Möbius group $\mathbb{P} S L_{2}(\mathbb{C})$ which abstractly are free groups of rank 2 .

Paper 4, Section I, F

commentFor every $k \in \mathbb{R}$, show that there is a closed bounded totally disconnected subset $X$ of some Euclidean space, such that $X$ has Hausdorff dimension at least $k$. [Standard properties of Hausdorff dimension may be quoted without proof if carefully stated.]

Paper 4, Section II, F

commentDefine three-dimensional hyperbolic space, the translation length of an isometry of hyperbolic 3 -space, and the axis of a hyperbolic isometry. Briefly explain how and why the latter two concepts are related.

Find the translation length of the isometries defined by (i) $z \mapsto k z, k \in \mathbb{C} \backslash\{0\}$ and (ii) $z \mapsto \frac{3 z+2}{7 z+5}$.

1.I.3G

commentProve that an isometry of Euclidean space $\mathbb{R}^{3}$ is an affine transformation.

Deduce that a finite group of isometries of $\mathbb{R}^{3}$ has a common fixed point.

1.II.11G

commentWhat is meant by an inversion in a circle in $\mathbb{C} \cup\{\infty\}$ ? Show that a composition of two inversions is a Möbius transformation.

Hence, or otherwise, show that if $C^{+}$and $C^{-}$are two disjoint circles in $\mathbb{C}$, then the composition of the inversions in $C^{+}$and $C^{-}$has two fixed points.

2.I.3G

commentState a theorem classifying lattices in $\mathbb{R}^{2}$. Define a frieze group.

Show there is a frieze group which is isomorphic to $\mathbb{Z}$ but is not generated by a translation, and draw a picture whose symmetries are this group.

3.I.3G

commentLet $\operatorname{dim}_{H}$ denote the Hausdorff dimension of a set in $\mathbb{R}^{n}$. Prove that if $\operatorname{dim}_{H}(F)<1$ then $F$ is totally disconnected.

[You may assume that if $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a Lipschitz map then

$\left.\operatorname{dim}_{H}(f(F)) \leqslant \operatorname{dim}_{H}(F) .\right]$

4.I $3 \mathrm{G}$

commentDefine the hyperbolic metric (in the sense of metric spaces) on the 3 -ball.

Given a finite set in hyperbolic 3 -space, show there is at least one closed ball of minimal radius containing that set.

4.II.12G

commentWhat does it mean for a subgroup $G$ of the Möbius group to be discrete?

Show that a discrete group necessarily acts properly discontinuously in hyperbolic 3-space.

[You may assume that a discrete subgroup of a matrix group is a closed subset.]

$1 . \mathrm{I} . 3 \mathrm{G}$

commentShow that there are two ways to embed a regular tetrahedron in a cube $C$ so that the vertices of the tetrahedron are also vertices of $C$. Show that the symmetry group of $C$ permutes these tetrahedra and deduce that the symmetry group of $C$ is isomorphic to the Cartesian product $S_{4} \times C_{2}$ of the symmetric group $S_{4}$ and the cyclic group $C_{2}$.

1.II.12G

commentDefine the Hausdorff $d$-dimensional measure $\mathcal{H}^{d}(C)$ and the Hausdorff dimension of a subset $C$ of $\mathbb{R}$.

Set $s=\log 2 / \log 3$. Define the Cantor set $C$ and show that its Hausdorff $s$-dimensional measure is at most $1 .$

Let $\left(X_{n}\right)$ be independent Bernoulli random variables that take the values 0 and 2 , each with probability $\frac{1}{2}$. Define

$\xi=\sum_{n=1}^{\infty} \frac{X_{n}}{3^{n}}$

Show that $\xi$ is a random variable that takes values in the Cantor set $C$.

Let $U$ be a subset of $\mathbb{R}$ with $3^{-(k+1)} \leqslant \operatorname{diam}(U)<3^{-k}$. Show that $\mathbb{P}(\xi \in U) \leqslant 2^{-k}$ and deduce that, for any set $U \subset \mathbb{R}$, we have

$\mathbb{P}(\xi \in U) \leqslant 2(\operatorname{diam}(U))^{s}$

Hence, or otherwise, prove that $\mathcal{H}^{s}(C) \geqslant \frac{1}{2}$ and that the Cantor set has Hausdorff dimension $s$.

2.I.3G

commentExplain what is meant by a lattice in the Euclidean plane $\mathbb{R}^{2}$. Prove that such a lattice is either $\mathbb{Z} \boldsymbol{w}$ for some vector $\boldsymbol{w} \in \mathbb{R}^{2}$ or else $\mathbb{Z} \boldsymbol{w}_{1}+\mathbb{Z} \boldsymbol{w}_{2}$ for two linearly independent vectors $\boldsymbol{w}_{1}, \boldsymbol{w}_{2}$ in $\mathbb{R}^{2}$.

3.I.3G

commentLet $G$ be a 2-dimensional Euclidean crystallographic group. Define the lattice and point group corresponding to $G$.

Prove that any non-trivial rotation in the point group of $G$ must have order $2,3,4$ or $6 .$

4.I $. 3 \mathrm{G}$

commentLet $\Gamma$ be a circle on the Riemann sphere. Explain what it means to say that two points of the sphere are inverse points for the circle $\Gamma$. Show that, for each point $z$ on the Riemann sphere, there is a unique point $z^{\prime}$ with $z, z^{\prime}$ inverse points. Define inversion in $\Gamma$.

Prove that the composition of an even number of inversions is a Möbius transformation.

4.II.12G

commentExplain what it means to say that a group $G$ is a Kleinian group. What is the definition of the limit set for the group $G$ ? Prove that a fixed point of a parabolic element in $G$ must lie in the limit set.

Show that the matrix $\left(\begin{array}{cc}1+a w & -a w^{2} \\ a & 1-a w\end{array}\right)$ represents a parabolic transformation for any non-zero choice of the complex numbers $a$ and $w$. Find its fixed point.

The Gaussian integers are $\mathbb{Z}[i]=\{m+i n: m, n \in \mathbb{Z}\}$. Let $G$ be the set of Möbius transformations $z \mapsto \frac{a z+b}{c z+d}$ with $a, b, c, d \in \mathbb{Z}[i]$ and $a d-b c=1$. Prove that $G$ is a Kleinian group. For each point $w=\frac{p+i q}{r}$ with $p, q, r$ non-zero integers, find a parabolic transformation $T \in G$ that fixes $w$. Deduce that the limit set for $G$ is all of the Riemann sphere.

1.I.3G

commentLet $G$ be a subgroup of the group of isometries $\operatorname{Isom}\left(\mathbb{R}^{2}\right)$ of the Euclidean plane. What does it mean to say that $G$ is discrete?

Supposing that $G$ is discrete, show that the subgroup $G_{T}$ of $G$ consisting of all translations in $G$ is generated by translations in at most two linearly independent vectors in $\mathbb{R}^{2}$. Show that there is a homomorphism $G \rightarrow O(2)$ with kernel $G_{T}$.

Draw, and briefly explain, pictures which illustrate two different possibilities for $G$ when $G_{T}$ is isomorphic to the additive group $\mathbb{Z}$.

1.II.12G

commentWhat is the limit set of a subgroup $G$ of Möbius transformations?

Suppose that $G$ is complicated and has no finite orbit in $\mathbb{C} \cup\{\infty\}$. Prove that the limit set of $G$ is infinite. Can the limit set be countable?

State Jørgensen's inequality, and deduce that not every two-generator subgroup $G=\langle A, B\rangle$ of Möbius transformations is discrete. Briefly describe two examples of discrete two-generator subgroups, one for which the limit set is connected and one for which it is disconnected.

2.I.3G

commentDescribe the geodesics in the disc model of the hyperbolic plane $\mathbb{H}^{2}$.

Define the area of a region in $\mathbb{H}^{2}$. Compute the area $A(r)$ of a hyperbolic circle of radius $r$ from the definition just given. Compute the circumference $C(r)$ of a hyperbolic circle of radius $r$, and check explicitly that $d A(r) / d r=C(r)$.

How could you define $\pi$ geometrically if you lived in $\mathbb{H}^{2}$ ? Briefly justify your answer.

3.I.3G

commentBy considering fixed points in $\mathbb{C} \cup\{\infty\}$, prove that any complex Möbius transformation is conjugate either to a map of the form $z \mapsto k z$ for some $k \in \mathbb{C}$ or to $z \mapsto z+1$. Deduce that two Möbius transformations $g, h$ (neither the identity) are conjugate if and only if $\operatorname{tr}^{2}(g)=\operatorname{tr}^{2}(h)$.

Does every Möbius transformation $g$ also have a fixed point in $\mathbb{H}^{3}$ ? Briefly justify your answer.

4.I.3G

commentShow that a set $F \subset \mathbb{R}^{n}$ with Hausdorff dimension strictly less than one is totally disconnected.

What does it mean for a Möbius transformation to pair two discs? By considering a pair of disjoint discs and a pair of tangent discs, or otherwise, explain in words why there is a 2-generator Schottky group with limit set $\Lambda \subset \mathbb{S}^{2}$ which has Hausdorff dimension at least 1 but which is not homeomorphic to a circle.

4.II.12G

commentFor real $s \geqslant 0$ and $F \subset \mathbb{R}^{n}$, give a careful definition of the $s$-dimensional Hausdorff measure of $F$ and of the Hausdorff dimension $\operatorname{dim}_{H}(F)$ of $F$.

For $1 \leqslant i \leqslant k$, suppose $S_{i}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a similarity with contraction factor $c_{i} \in(0,1)$. Prove there is a unique non-empty compact invariant set $I$ for the $\left\{S_{i}\right\}$. State a formula for the Hausdorff dimension of $I$, under an assumption on the $S_{i}$ you should state.

Hence show the Hausdorff dimension of the fractal $F$ given by iterating the scheme below (at each stage replacing each edge by a new copy of the generating template) is $\operatorname{dim}_{H}(F)=3 / 2$.

[Numbers denote lengths]