Geometry And Groups

# Geometry And Groups

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Paper 1, Section I, F

commentLet $G \leqslant S O(3)$ be a finite group. Suppose $G$ does not preserve any plane in $\mathbb{R}^{3}$. Show that for any point $p$ in the unit sphere $S^{2} \subset \mathbb{R}^{3}$, the stabiliser $\operatorname{Stab}_{G}(p)$ contains at most 5 elements.

Paper 1, Section II, F

commentProve that an orientation-preserving isometry of the ball-model of hyperbolic space $\mathbb{H}^{3}$ which fixes the origin is an element of $S O(3)$. Hence, or otherwise, prove that a finite subgroup of the group of orientation-preserving isometries of hyperbolic space $\mathbb{H}^{3}$ has a common fixed point.

Can an infinite non-cyclic subgroup of the isometry group of $\mathbb{H}^{3}$ have a common fixed point? Can any such group be a Kleinian group? Justify your answers.

Paper 2, Section I, F

commentLet $g, h$ be non-identity Möbius transformations. Prove that $g$ and $h$ commute if and only if one of the following holds:

$\operatorname{Fix}(g)=\operatorname{Fix}(h)$;

$g, h$ are involutions each of which exchanges the other's fixed points.

Give an example to show that the second case can occur.

Paper 3, Section I, F

commentLet $\mathbb{H}^{2}$ denote the hyperbolic plane, and $T \subset \mathbb{H}^{2}$ be a non-degenerate triangle, i.e. the bounded region enclosed by three finite-length geodesic arcs. Prove that the three angle bisectors of $T$ meet at a point.

Must the three vertices of $T$ lie on a hyperbolic circle? Justify your answer.

Paper 4, Section I, F

commentDefine the limit set $\Lambda(G)$ of a Kleinian group $G$. Assuming that $G$ has no finite orbit in $\mathbb{H}^{3} \cup S_{\infty}^{2}$, and that $\Lambda(G) \neq \emptyset$, prove that if $E \subset \mathbb{C} \cup\{\infty\}$ is any non-empty closed set which is invariant under $G$, then $\Lambda(G) \subset E$.

Paper 4, Section II, F

commentDefine the $s$-dimensional Hausdorff measure $\mathcal{H}^{s}(F)$ of a set $F \subset \mathbb{R}^{N}$. Explain briefly how properties of this measure may be used to define the Hausdorff dimension $\operatorname{dim}_{H}(F)$ of such a set.

Prove that the limit sets of conjugate Kleinian groups have equal Hausdorff dimension. Hence, or otherwise, prove that there is no subgroup of $\mathbb{P} S L(2, \mathbb{R})$ which is conjugate in $\mathbb{P} S L(2, \mathbb{C})$ to $\mathbb{P} S L(2, \mathbb{Z} \oplus \mathbb{Z} i)$.

Paper 1, Section I, G

commentShow that any pair of lines in hyperbolic 3-space that does not have a common endpoint must have a common normal. Is this still true when the pair of lines does have a common endpoint?

Paper 1, Section II, G

commentDefine the modular group $\Gamma$ acting on the upper half-plane.

Describe the set $S$ of points $z$ in the upper half-plane that have $\operatorname{Im}(T(z)) \leqslant \operatorname{Im}(z)$ for each $T \in \Gamma$. Hence find a fundamental set for $\Gamma$ acting on the upper half-plane.

Let $A$ and $J$ be the two Möbius transformations

$A: z \mapsto z+1 \quad \text { and } \quad J: z \mapsto-1 / z$

When is $\operatorname{Im}(J(z))>\operatorname{Im}(z) ?$

For any point $z$ in the upper half-plane, show that either $z \in S$ or else there is an integer $k$ with

$\operatorname{Im}\left(J\left(A^{k}(z)\right)\right)>\operatorname{Im}(z)$

Deduce that the modular group is generated by $A$ and $J$.

Paper 2, Section I, G

commentLet $\ell_{1}, \ell_{2}$ be two straight lines in Euclidean 3-space. Show that there is a rotation about some axis through an angle $\pi$ that maps $\ell_{1}$ onto $\ell_{2}$. Is this rotation unique?

Paper 3, Section I, $3 G$

commentLet $\Lambda$ be a rank 2 lattice in the Euclidean plane. Show that the group $G$ of all Euclidean isometries of the plane that map $\Lambda$ onto itself is a discrete group. List the possible sizes of the point groups for $G$ and give examples to show that point groups of these sizes do arise.

[You may quote any standard results without proof.]

Paper 4, Section I, $3 G$

commentLet $\Delta_{1}, \Delta_{2}$ be two disjoint closed discs in the Riemann sphere with bounding circles $\Gamma_{1}, \Gamma_{2}$ respectively. Let $J_{k}$ be inversion in the circle $\Gamma_{k}$ and let $T$ be the Möbius transformation $J_{2} \circ J_{1}$.

Show that, if $w \notin \Delta_{1}$, then $T(w) \in \Delta_{2}$ and so $T^{n}(w) \in \Delta_{2}$ for $n=1,2,3, \ldots$ Deduce that $T$ has a fixed point in $\Delta_{2}$ and a second in $\Delta_{1}$.

Deduce that there is a Möbius transformation $A$ with

$A\left(\Delta_{1}\right)=\{z:|z| \leqslant 1\} \quad \text { and } \quad A\left(\Delta_{2}\right)=\{z:|z| \geqslant R\}$

for some $R>1$.

Paper 4, Section II, $12 \mathrm{G}$

commentDefine the limit set for a Kleinian group. If your definition of the limit set requires an arbitrary choice of a base point, you should prove that the limit set does not depend on this choice.

Let $\Delta_{1}, \Delta_{2}, \Delta_{3}, \Delta_{4}$ be the four discs $\{z \in \mathbb{C}:|z-c| \leqslant 1\}$ where $c$ is the point $1+i, 1-i,-1-i,-1+i$ respectively. Show that there is a parabolic Möbius transformation $A$ that maps the interior of $\Delta_{1}$ onto the exterior of $\Delta_{2}$ and fixes the point where $\Delta_{1}$ and $\Delta_{2}$ touch. Show further that we can choose $A$ so that it maps the unit disc onto itself.

Let $B$ be the similar parabolic transformation that maps the interior of $\Delta_{3}$ onto the exterior of $\Delta_{4}$, fixes the point where $\Delta_{3}$ and $\Delta_{4}$ touch, and maps the unit disc onto itself. Explain why the group generated by $A$ and $B$ is a Kleinian group $G$. Find the limit set for the group $G$ and justify your answer.

Paper 1, Section I, G

commentLet $G$ be a crystallographic group of the Euclidean plane. Define the lattice and the point group of $G$. Suppose that the lattice for $G$ is $\{(k, 0): k \in \mathbb{Z}\}$. Show that there are five different possibilities for the point group. Show that at least one of these point groups can arise from two groups $G$ that are not conjugate in the group of all isometries of the Euclidean plane.

Paper 1, Section II, G

commentDefine the axis of a loxodromic Möbius transformation acting on hyperbolic 3-space.

When do two loxodromic transformations commute? Justify your answer.

Let $G$ be a Kleinian group that contains a loxodromic transformation. Show that the fixed point of any loxodromic transformation in $G$ lies in the limit set of $G$. Prove that the set of such fixed points is dense in the limit set. Give examples to show that the set of such fixed points can be equal to the limit set or a proper subset.

Paper 2, Section I, G

commentDefine the modular group acting on the upper half-plane. Explain briefly why it acts discontinuously and describe a fundamental domain. You should prove that the region which you describe is a fundamental domain.

Paper 3, Section I, G

commentLet $A$ be a Möbius transformation acting on the Riemann sphere. Show that, if $A$ is not loxodromic, then there is a disc $\Delta$ in the Riemann sphere with $A(\Delta)=\Delta$. Describe all such discs for each Möbius transformation $A$.

Hence, or otherwise, show that the group $G$ of Möbius transformations generated by

$A: z \mapsto i z \quad \text { and } \quad B: z \mapsto 2 z$

does not map any disc onto itself.

Describe the set of points of the Riemann sphere at which $G$ acts discontinuously. What is the quotient of this set by the action of $G$ ?

Paper 4, Section I, G

commentExplain briefly how to extend a Möbius transformation

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

from the boundary of the upper half-space $\mathbb{R}_{+}^{3}$ to give a hyperbolic isometry $\widetilde{T}$of the upper half-space. Write down explicitly the extension of the transformation $z \mapsto \lambda^{2} z$ for any constant $\lambda \in \mathbb{C} \backslash\{0\}$.

Show that, if $\tilde{T}$ has an axis, which is a hyperbolic line that is mapped onto itself by $\tilde{T}$ with the orientation preserved, then $\widetilde{T}$moves each point of this axis by the same hyperbolic distance, $\ell$ say. Prove that

$\ell=2|\log | \frac{1}{2}\left(a+d+\sqrt{(a+d)^{2}-4}\right)||$

Paper 4, Section II, $12 \mathrm{G}$

commentDefine the Hausdorff dimension of a subset of the Euclidean plane.

Let $\Delta$ be a closed disc of radius $r_{0}$ in the Euclidean plane. Define a sequence of sets $K_{n} \subseteq \Delta, n=1,2, \ldots$, as follows: $K_{1}=\Delta$ and for each $n \geqslant 1$ a subset $K_{n+1} \subset K_{n}$ is produced by replacing each component disc $\Gamma$ of $K_{n}$ by three disjoint, closed discs inside $\Gamma$ with radius at most $c_{n}$ times the radius of $\Gamma$. Let $K$ be the intersection of the sets $K_{n}$. Show that if the factors $c_{n}$ converge to a limit $c$ with $0<c<1$, then the Hausdorff dimension of $K$ is at most $\log \frac{1}{3} / \log c$.

Paper 1, Section I, G

commentLet $G$ be a finite subgroup of $\mathrm{SO}(3)$ and let $\Omega$ be the set of unit vectors that are fixed by some non-identity element of $G$. Show that the group $G$ permutes the unit vectors in $\Omega$ and that $\Omega$ has at most three orbits. Describe these orbits when $G$ is the group of orientation-preserving symmetries of a regular dodecahedron.

Paper 1, Section II, G

commentProve that a group of Möbius transformations is discrete if, and only if, it acts discontinuously on hyperbolic 3 -space.

Let $G$ be the set of Möbius transformations $z \mapsto \frac{a z+b}{c z+d}$ with

$a, b, c, d \in \mathbb{Z}[i]=\{u+i v: u, v \in \mathbb{Z}\} \quad \text { and } \quad a d-b c=1$

Show that $G$ is a group and that it acts discontinuously on hyperbolic 3-space. Show that $G$ contains transformations that are elliptic, parabolic, hyperbolic and loxodromic.

Paper 2, Section $I$, $3 \mathrm{G}$

commentLet $A$ and $B$ be two rotations of the Euclidean plane $\mathbb{E}^{2}$ about centres $a$ and $b$ respectively. Show that the conjugate $A B A^{-1}$ is also a rotation and find its fixed point. When do $A$ and $B$ commute? Show that the commutator $A B A^{-1} B^{-1}$ is a translation.

Deduce that any group of orientation-preserving isometries of the Euclidean plane either fixes a point or is infinite.

Paper 3, Section I, $3 G$

commentDefine a Kleinian group.

Give an example of a Kleinian group that is a free group on two generators and explain why it has this property.

Paper 4, Section I, G

commentDefine inversion in a circle $\Gamma$ on the Riemann sphere. You should show from your definition that inversion in $\Gamma$ exists and is unique.

Prove that the composition of an even number of inversions is a Möbius transformation of the Riemann sphere and that every Möbius transformation is the composition of an even number of inversions.

Paper 4, Section II, G

commentDefine a lattice in $\mathbb{R}^{2}$ and the rank of such a lattice.

Let $\Lambda$ be a rank 2 lattice in $\mathbb{R}^{2}$. Choose a vector $\boldsymbol{w}_{1} \in \Lambda \backslash\{\boldsymbol{0}\}$ with $\left\|\boldsymbol{w}_{1}\right\|$ as small as possible. Then choose $\boldsymbol{w}_{2} \in \Lambda \backslash \mathbb{Z} \boldsymbol{w}_{1}$ with $\left\|\boldsymbol{w}_{2}\right\|$ as small as possible. Show that $\Lambda=\mathbb{Z} \boldsymbol{w}_{1}+\mathbb{Z} \boldsymbol{w}_{2}$.

Suppose that $\boldsymbol{w}_{1}$ is the unit vector $\left(\begin{array}{l}1 \\ 0\end{array}\right)$. Draw the region of possible values for $\boldsymbol{w}_{2}$. Suppose that $\Lambda$ also equals $\mathbb{Z} \boldsymbol{v}_{1}+\mathbb{Z} \boldsymbol{v}_{2}$. Prove that

$\boldsymbol{v}_{1}=a \boldsymbol{w}_{1}+b \boldsymbol{w}_{2} \quad \text { and } \quad \boldsymbol{v}_{2}=c \boldsymbol{w}_{1}+d \boldsymbol{w}_{2}$

for some integers $a, b, c, d$ with $a d-b c=\pm 1$.

1.I $. 3 \mathrm{~F} \quad$

commentSuppose $S_{i}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a similarity with contraction factor $c_{i} \in(0,1)$ for $1 \leqslant i \leqslant k$. Let $X$ be the unique non-empty compact invariant set for the $S_{i}$ 's. State a formula for the Hausdorff dimension of $X$, under an assumption on the $S_{i}$ 's you should state. Hence compute the Hausdorff dimension of the subset $X$ of the square $[0,1]^{2}$ defined by dividing the square into a $5 \times 5$ array of squares, removing the open middle square $(2 / 5,3 / 5)^{2}$, then removing the middle $1 / 25$ th of each of the remaining 24 squares, and so on.

1.II.12F

commentCompute the area of the ball of radius $r$ around a point in the hyperbolic plane. Deduce that, for any tessellation of the hyperbolic plane by congruent, compact tiles, the number of tiles which are at most $n$ "steps" away from a given tile grows exponentially in $n$. Give an explicit example of a tessellation of the hyperbolic plane.

2.I.3F

commentDetermine whether the following elements of $\operatorname{PSL}_{2}(\mathbb{R})$ are elliptic, parabolic, or hyperbolic. Justify your answers.

$\left(\begin{array}{cc} 5 & 8 \\ -2 & -3 \end{array}\right), \quad\left(\begin{array}{cc} -3 & 1 \\ 2 & -1 \end{array}\right)$

In the case of the first of these transformations find the fixed points.

3.I.3F

commentLet $G$ be a discrete subgroup of the Möbius group. Define the limit set of $G$ in $S^{2}$. If $G$ contains two loxodromic elements whose fixed point sets in $S^{2}$ are different, show that the limit set of $G$ contains no isolated points.

4.I.3F

commentWhat is a crystallographic group in the Euclidean plane? Prove that, if $G$ is crystallographic and $g$ is a nontrivial rotation in $G$, then $g$ has order $2,3,4$, or 6 .

4.II.12F

commentLet $G$ be a discrete subgroup of $\operatorname{PSL}_{2}(\mathbb{C})$. Show that $G$ is countable. Let $G=\left\{g_{1}, g_{2}, \ldots\right\}$ be some enumeration of the elements of $G$. Show that for any point $p$ in hyperbolic 3-space $\mathbb{H}^{3}$, the distance $d_{h y p}\left(p, g_{n}(p)\right)$ tends to infinity. Deduce that a subgroup $G$ of $\mathrm{PSL}_{2}(\mathbb{C})$ is discrete if and only if it acts properly discontinuously on $\mathbb{H}^{3}$.