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Theorem 3.2.6. The points \( x \) and \( y \) are inverse points with respect to the sphere \( \sum \) if and only if every sphere through \( x \) and \( y \) is orthogonal to \( \sum \) . | Proof. This is clearly true when \( \sum \) is a plane: it is true in general by the invariance of both inverse points and orthogonality. | No |
Theorem 3.2.7. A map \( \phi : {\widehat{\mathbb{R}}}^{n} \rightarrow {\widehat{\mathbb{R}}}^{n} \) is a Möbius transformation if and only if it preserves cross-ratios. | Proof. As each Möbius map that changes Euclidean distance by a constant factor leaves the expression (3.2.6) invariant, it is only necessary to consider the map \( x \mapsto {x}^{ * } \) . As (see (3.1.5))\n\n\[ \left| {{x}^{ * } - {y}^{ * }}\right| = \frac{\left| x - y\right| }{\left| x\right| \left| y\right| } \]\n\n... | Yes |
Theorem 3.4.1. Let \( \phi \) be a Möbius transformation with \( \phi \left( 0\right) = 0 \) and \( \phi \left( {B}^{n}\right) = {B}^{n} \). Then \( \phi \left( x\right) = {xA} \) for some orthogonal matrix \( A \). | Proof. By Theorem 3.2.5, \( \phi \) fixes \( \infty \) and, as in the proof of Theorem 3.2.7, we see that \( \phi \) is a Euclidean similarity. Because \( \phi \) fixes the origin and leaves \( {S}^{n - 1} \) invariant, it is actually a Euclidean isometry. The result now follows from Theorem 3.1.4. | Yes |
Theorem 3.4.2. Let \( \phi \) be the reflection in \( S\left( {a, r}\right) \) . Then the following are equivalent:\n\n(i) \( S\left( {a, r}\right) \) and \( {S}^{n - 1} \) are orthogonal;\n\n(ii) \( \phi \left( {a}^{ * }\right) = 0 \) (equivalently, \( \phi \left( 0\right) = {a}^{ * } \) );\n\n(iii) \( \phi \left( {B}... | Proof. As\n\n\[ \phi \left( 0\right) = a - {r}^{2}{a}^{ * } \]\n\n\[ = \left( {{\left| a\right| }^{2} - {r}^{2}}\right) {a}^{ * } \]\n\nwe see that (i) and (ii) are equivalent. The assertion that (iii) implies (ii) is simply the fact that \( a \) and \( {a}^{ * } \) map to inverse points with respect to \( {S}^{n - 1} ... | Yes |
Theorem 3.5.1. Let \( \phi \) be a Möbius transformation.\n\n(i) If \( \phi \left( {B}^{n}\right) = {B}^{n} \) then\n\n\[ \phi \left( x\right) = \left( {\sigma x}\right) A \]\n\nwhere \( \sigma \) is a reflection in some sphere orthogonal to \( {S}^{n - 1} \) and \( A \) is an orthogonal matrix. | Proof. If \( \phi \) preserves \( {B}^{n} \), let \( \sigma \) be the reflection in the sphere \( S\left( {a, r}\right) \) where \( a = {\phi }^{-1}\left( \infty \right) \) and \( {\left| a\right| }^{2} = 1 + {r}^{2} \) . By Theorem 3.4.2, \( \sigma \) (and hence \( {\phi \sigma } \) ) preserves \( {B}^{n} \) . By comp... | Yes |
Theorem 3.6.2. Let \( D \) be a subdomain of \( {\widehat{\mathbb{R}}}^{n} \) and suppose that \( \xi \) and \( \zeta \) are distinct points in \( {\widehat{\mathbb{R}}}^{n} \) . If \( \phi \) in \( \operatorname{GM}\left( {\widehat{\mathbb{R}}}^{n}\right) \) does not assume the values \( \xi \) and \( \zeta \) in \( D... | Proof of Theorem 3.6.2. Suppose that \( x \) and \( y \) are distinct points in \( D \) and that \( \alpha \) and \( \beta \) are distinct points outside of \( D \) . By Theorem 3.2.7, the product\n\n\[ \left\lbrack {x,\alpha, y,\beta }\right\rbrack \cdot \left\lbrack {x,\beta, y,\alpha }\right\rbrack \]\nof cross-rati... | Yes |
Proposition 3.6.3. A family \( \mathcal{F} \) of Möbius transformations of \( \left( {{\widehat{\mathbb{R}}}^{n}, d}\right) \) onto itself is normal in a subdomain \( D \) of \( {\widehat{\mathbb{R}}}^{n} \) if it is equicontinuous on every compact subset of \( D \) . | Proof. We only sketch the proof as the interested reader can find a proof of the Arzela-Ascoli Theorem elsewhere in the literature. Find a sequence \( {x}_{1},{x}_{2},\ldots \) which is dense in \( D \) . Given a sequence \( {\phi }_{1},{\phi }_{2},\ldots \) in \( \mathcal{F} \) we can find (because \( {\widehat{\mathb... | Yes |
Theorem 3.6.4. Let \( D \) be a subdomain of \( {\widehat{\mathbb{R}}}^{n} \) and let \( \mathcal{F} \) be a family of Möbius transformations. Suppose that for every \( \phi \) in \( \mathcal{F} \), there are two points \( {\alpha }_{\phi },{\beta }_{\phi } \) in \( {\widehat{\mathbb{R}}}^{n} \) which are not taken as ... | Proof. We simply apply Theorem 3.6.2 with \( \xi = {\alpha }_{\phi },\zeta = {\beta }_{\phi } \) and we find that \( \mathcal{F} \) is equicontinuous (in fact, it satisfies a uniform Lipschitz condition) on every compact subset of \( D \) . | Yes |
Theorem 3.6.5. Let \( {\phi }_{1},{\phi }_{2},\ldots \) be Möbius transformations and suppose that \( {\phi }_{n}\left( {x}_{j}\right) \rightarrow {y}_{j} \) for three distinct points \( {x}_{1},{x}_{2},{x}_{3} \) and three distinct points \( {y}_{1},{y}_{2},{y}_{3} \) . Then \( {\phi }_{1},{\phi }_{2},\ldots \) contai... | Proof. By the deletion of a finite number of the \( {\phi }_{j} \) (which clearly does not affect the result) we may assume that for each \( n, i \) and \( j\left( {i \neq j}\right) \) we have\n\n\[ d\left( {{\phi }_{n}{x}_{i},{\phi }_{n}{x}_{j}}\right) \geq \frac{1}{2}d\left( {{y}_{i},{y}_{j}}\right) > 0. \]\n\nIt fol... | Yes |
Theorem 3.7.1. \( \operatorname{GM}\left( {\widehat{\mathbb{R}}}^{n}\right) \) is a topological group with respect to the topology induced by the metric \( D \) . | Proof. From Theorem 3.6.1, we see that for each \( \phi \) in \( \operatorname{GM}\left( {\widehat{\mathbb{R}}}^{n}\right) \) there is a positive constant \( c\left( \phi \right) \) such that for all \( x \) and \( y \) we have\n\n\[ d\left( {{\phi x},{\phi y}}\right) \leq c\left( \phi \right) d\left( {x, y}\right) . \... | Yes |
Lemma 3.7.2. (i) The map \( \phi \mapsto \phi \left( 0\right) \) of \( \operatorname{GM}\left( {B}^{n + 1}\right) \) onto \( {B}^{n + 1} \) is continuous | To prove (i) we suppose first that \( D\left( {{\phi }_{n}, I}\right) < \varepsilon \) . Each Euclidean diameter \( {L}_{j} \) of \( {B}^{n + 1} \) is mapped by \( {\phi }_{n} \) to a circular arc \( {\phi }_{n}\left( {L}_{j}\right) \) (orthogonal to \( \left. {S}^{n}\right) \) in \( {B}^{n + 1} \) whose end-points are... | Yes |
Theorem 3.7.3. The bijection \( \phi \mapsto \left( {{A}_{\phi }, a}\right) \) is a homeomorphism of \( \operatorname{GM}\left( {B}^{n + 1}\right) \) onto \( \mathrm{O}\left( {n + 1}\right) \times {B}^{n + 1} \) . | Proof. The proof consists of repeated applications of Theorem 3.7.1 and Lemma 3.7.2. First, \( a \mapsto {T}_{a} \) is continuous, hence so is the map \( \left( {{A}_{\phi }, a}\right) \mapsto \) \( \left( {{A}_{\phi },{T}_{a}}\right) \) . Also the map of \( \left( {{A}_{\phi },{T}_{a}}\right) \) into their composition... | Yes |
Theorem 3.7.5. The map\n\n\\[ \nF : \\left( {{x}_{0},\\ldots ,{x}_{n}}\\right) \\mapsto \\left( {\\frac{{x}_{1}}{1 + {x}_{0}},\\ldots ,\\frac{{x}_{n}}{1 + {x}_{0}}}\\right) \n\\]\n\n is an isometry of \\( Q \\) with the metric (3.7.1) onto \\( {B}^{n} \\) with the metric (3.7.2). | Proof. For brevity, we write\n\n\\[ \n\\left( {{y}_{1},\\ldots ,{y}_{n}}\\right) = \\left( {\\frac{{x}_{1}}{1 + {x}_{0}},\\ldots ,\\frac{{x}_{n}}{1 + {x}_{0}}}\\right) \n\\]\n\n and denote the vectors by \\( x \\) and \\( y \\) in the obvious way. As \\( x \\in Q \\), a computation yields\n\n\\[ \n{\\left| y\\right| }^... | Yes |
Theorem 3.7.6. The isometries of \( Q \) are precisely the \( \left( {n + 1}\right) \times \left( {n + 1}\right) \) matrices which preserve both the quadratic form \( q\left( {x, x}\right) \) and the half-space given by \( {x}_{0} > 0 \) . | Proof. First, let \( A \) be any matrix with the prescribed properties. As \( {x}_{0} > 0 \) is preserved and as\n\n\[ q\left( {{xA},{xA}}\right) = q\left( {x, x}\right) = 1, \]\n\nwhen \( x \in Q \) we see that \( A \) preserves \( Q \) . Moreover, for any curve \( \gamma \) on \( Q \), let \( \Gamma = {\gamma A} \) .... | Yes |
Theorem 3.7.7. \( \operatorname{GM}\left( {\widehat{\mathbb{R}}}^{n}\right) \) with the topology of uniform convergence in the chordal metric is isomorphic as a topological group to the group \( {\mathrm{O}}^{ + }\left( {1, n + 1}\right) \) of matrices. | In particular, if we identify \( {\widehat{\mathbb{R}}}^{2} \) with the extended complex plane, then \( \mathbf{M}\left( {\widehat{\mathbb{R}}}^{2}\right) \) is (as we shall see) the class of complex Möbius transformations\n\n\[ z \mapsto \frac{{az} + b}{{cz} + d},\;{ad} - {bc} \neq 0, \]\n\nand this is isomorphic to t... | No |
Theorem 4.2.1. For each \( g \) in \( \mathcal{M} \), we have\n\n\[ \parallel g{\parallel }^{2} = 2\cosh \rho \left( {j,{gj}}\right) . \] | Proof. Write\n\n\[ g\left( z\right) = \frac{{az} + b}{{cz} + d},\;{ad} - {bc} = 1 \]\n\nthen by (4.1.4) (with \( z = 0 \) and \( t = 1 \) ),\n\n\[ g\left( j\right) = \frac{\left( {b\bar{d} + a\bar{c}}\right) + j}{{\left| c\right| }^{2} + {\left| d\right| }^{2}}. \]\n\nAccording to (3.3.4), if \( {\zeta }_{1} = {z}_{1} ... | Yes |
Theorem 4.2.2. The following statements are equivalent.\n\n(i) \( A \in \mathrm{{SU}}\left( {2,\mathbb{C}}\right) \) ;\n\n(ii) \( g\left( j\right) = j \) ;\n\n(iii) \( \parallel g{\parallel }^{2} = 2 \) ;\n\n(iv) \( {fg}{f}^{-1} \) is a linear orthogonal transformation;\n\n(v) \( g \) is an isometry of the chordal metr... | Proof. The equivalence of (ii) and (iii) is a direct corollary of Theorem 4.2.1. As \( A \in \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) we have \( \parallel A{\parallel }^{2} = \parallel g{\parallel }^{2} \) and the equivalence of (i) and (iii) is a direct consequence of Theorem 2.5.1.\n\nNext, (ii) is equivalent to\... | Yes |
Theorem 4.3.1. Let \( f \) and \( g \) be Möbius transformations, neither the identity. Then \( f \) and \( g \) are conjugate if and only if \( {\operatorname{tr}}^{2}\left( f\right) = {\operatorname{tr}}^{2}\left( g\right) \) . | Proof. We have already noted (following (4.2.1)) that if \( f \sim g \) then \( {\operatorname{tr}}^{2}\left( f\right) = {\operatorname{tr}}^{2}\left( g\right) \n\nNow assume that \( {\operatorname{tr}}^{2}\left( f\right) = {\operatorname{tr}}^{2}\left( g\right) \) . We know that \( f \) and \( g \) are each conjugate ... | Yes |
Theorem 4.3.6. Let \( g \) and \( h \) be Möbius transformations other than I. The following statements are equivalent:\n\n(i) \( {hg} = {gh} \) ;\n\n(ii) \( h\left( {F}_{g}\right) = {F}_{g}, g\left( {F}_{h}\right) = {F}_{h} \) ;\n\n(iii) either \( {F}_{g} = {F}_{h} \) or \( g \) and \( h \) have a common fixed point i... | Proof. First, (4.3.1) shows that (i) implies (ii).\n\nThe proof that (iii) implies (i) is easy. If \( {F}_{g} = {F}_{h} \) then \( g \) and \( h \) have a common fixed point and so by Theorem 4.3.5, \( \left\lbrack {g, h}\right\rbrack = I \) : thus in this case, \( {gh} = {hg} \) . The other alternative offered by (iii... | Yes |
Theorem 4.3.7. A subgroup \( G \) of \( \mathcal{M} \) contains only elliptic elements (and \( I \) ) if and only if the elements of \( G \) have a common fixed point in \( {H}^{3} \) . | It follows from Definition 4.3.2 that if \( g\left( { \neq I}\right) \) is of finite order then \( g \) is necessarily elliptic. As every element in a finite group has finite order we have the following corollary.\n\nCorollary. The elements in a finite subgroup of \( \mathcal{M} \) have a common fixed point in \( {\mat... | No |
Lemma 4.3.8. Suppose that \( g, h \) and \( {gh} \) are elliptic. Then the fixed points of \( g \) and \( h \) in \( \widehat{\mathbb{C}} \) are concyclic. If, in addition, \( \left\lbrack {g, h}\right\rbrack \) is elliptic or I, then the axes \( {A}_{g} \) and \( {A}_{h} \) are concurrent in \( {H}^{3} \) . | Proof. If \( g \) and \( h \) have a common fixed point in \( \widehat{\mathbb{C}} \), then \( {F}_{g} \cup {F}_{h} \) has at most three points and so lies in some circle. If, in addition, \( \left\lbrack {g, h}\right\rbrack \) is elliptic or \( I \) , then from Theorem 4.3.5, \( {F}_{g} = {F}_{h} \) and so \( {A}_{g} ... | Yes |
Lemma 4.3.9. Let \( g \) and \( h \) be Möbius transformations \( \left( { \neq I}\right) \) which preserve \( {B}^{3} \) and fix the origin. Then\n\n(i) the elements of \( \langle g, h\rangle \) have the same axis and same fixed points or\n\n(ii) there is some fin \( \langle g, h\rangle \) such that the three axes \( ... | Proof of Lemma 4.3.9. Every element of \( \langle g, h\rangle \) fixes the origin and so is elliptic or \( I \) . For each such elliptic \( f \), let \( {A}_{f} \) denote the axis (of fixed points) of \( f \) in \( {B}^{3} \) . Note that by assumption, \( {A}_{g} \) and \( {A}_{h} \) are Euclidean diameters of \( {B}^{... | Yes |
Theorem 4.4.1. The map \( \theta : {S}_{4} \rightarrow {\mathcal{M}}_{0} \) is a homomorphism of \( {S}_{4} \) onto \( {\mathcal{M}}_{0} \) with kernel K. | Proof. Theorem 4.1.1. implies that \( {\mathcal{M}}_{0} \) has exactly six elements: these are the functions\n\n\[ \lambda ,1 - \lambda ,\lambda /\left( {\lambda - 1}\right) ,1/\lambda ,1/\left( {1 - \lambda }\right) ,\left( {\lambda - 1}\right) /\lambda \]\n\nof \( \lambda \) . There are six permutations \( \sigma \) ... | Yes |
Theorem 4.5.1. The topology \( \mathcal{T} \) induced on \( \mathcal{M} \) by \( \Phi \) coincides with the topology \( {\mathcal{T}}^{ * } \) of uniform convergence on \( \widehat{\mathbb{C}} \) . | Proof. It is sufficient to show that the map\n\n\[ \Phi : \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \rightarrow \left( {\mathcal{M},{\mathcal{T}}^{ * }}\right) \]\n\n(4.5.1)\n\nis open and continuous: see Proposition 1.4.1.\n\nAssuming that this has been established, observe that if \( X \) is in \( \mathrm{{SL}}\left(... | No |
Proposition 4.5.3. If \( A \) in \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) represents \( g \), then\n\n\[ \sigma \left( {g, I}\right) \leq \sqrt{6}\parallel A - I\parallel \] | Proof of Proposition 4.5.3. There is a unitary matrix \( B \) representing a Möbius map \( h \) such that \( {hg}{h}^{-1} \) fixes \( \infty \) ( \( h \) corresponds to a rotation of the sphere moving a selected fixed point of \( g \) to \( \infty \) ). By Theorems 2.5.2 and 4.2.2 we have\n\n\[ \parallel A - I\parallel... | Yes |
Proposition 4.5.4. Let \( {g}_{1},{g}_{2},\ldots \) be Möbius transformations and suppose that \( {g}_{n}\left( w\right) \rightarrow w \) for \( w = 0,1,\infty \) . Then:\n\n(i) there exist matrices \( {A}_{n} \) representing \( {g}_{n} \) which converge to \( I \) ; and\n\n(ii) \( {g}_{n} \rightarrow I \) uniformly on... | Proof. Choose matrices\n\n\[ \n{A}_{n} = {\varepsilon }_{n}\left( \begin{array}{ll} {a}_{n} & {b}_{n} \\ {c}_{n} & {d}_{n} \end{array}\right) \n\]\n\nin \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) representing \( {g}_{n} \) where \( {\varepsilon }_{n} \) is 1 or -1 and is to be chosen later. In the following argume... | Yes |
Theorem 4.5.5. Suppose that \( K \) is a compact subset of a domain \( D \) in \( \widehat{\mathbb{C}} \) and that \( g \) omits the values 0 and \( \infty \) in \( D \) . Then for some positive \( m \) depending only on \( D \) and \( K \), we have\n\n\[ d\left( {{gz},{gw}}\right) \leq \frac{{md}\left( {z, w}\right) }... | Proof. Define \( {m}_{1} \) by\n\n\[ 2{m}_{1} = \inf \{ d\left( {z, w}\right) : z \in K, w \notin D\} \]\n\nand suppose that\n\n\[ g\left( z\right) = \frac{{az} + b}{{cz} + d},\;{ad} - {bc} = 1. \]\n\nAs \( {g}^{-1}\left( \infty \right) \notin D \), we see that for \( z \) in \( K \) ,\n\n\[ 2{m}_{1} \leq d\left( {z,{g... | Yes |
Theorem 5.1.2. Let \( g \) be loxodromic and suppose that \( f \) and \( g \) have exactly one fixed point in common. Then \( \langle f, g\rangle \) is not discrete. | Proof. As discreteness is preserved under conjugation we may assume that the common fixed point is \( \infty \) and, say,\n\n\[ g\left( z\right) = {\alpha z}\;\left( {\left| \alpha \right| > 1}\right) ,\;f\left( z\right) = {az} + b \]\n\n(if necessary, we may replace \( g \) by \( {g}^{-1} \) ).\n\nThen\n\n\[ {g}^{-n}f... | Yes |
Theorem 5.1.3. Every non-elementary subgroup \( G \) of \( \mathcal{M} \) contains infinitely many loxodromic elements, no two of which have a common fixed point. | Proof. We begin by showing that \( G \) has some loxodromic elements. Suppose, then, that \( G \) has no loxodromic elements. If \( G \) contains only \( I \) and elliptic elements then \( G \) is elementary. It follows that \( G \) contains a parabolic element which we may take to be\n\n\[ f\left( z\right) = z + 1 \]\... | Yes |
Theorem 5.1.4. Let \( f\left( { \neq I}\right) \) be a Möbius transformation not of order two and define the map \( \theta : \mathcal{M} \rightarrow \mathcal{M} \) by \( \theta \left( g\right) = {gf}{g}^{-1} \) . If for some \( n \), we have \( {\theta }^{n}\left( g\right) = f \) , then \( \langle f, g\rangle \) is ele... | Proof. Define \( {g}_{0} = g \) and \( {g}_{n} = {\theta }^{n}\left( g\right) \) so for \( m \geq 0 \), \[ {g}_{m + 1} = {g}_{m}f{\left( {g}_{m}\right) }^{-1}. \] Suppose first that \( f \) is parabolic; then without loss of generality, \( f\left( z\right) = z + 1 \) . As \( {g}_{1},\ldots ,{g}_{n} \) are conjugate to ... | Yes |
Theorem 5.3.2. A subgroup \( G \) of \( \mathcal{M} \) is discrete if and only if it acts discontinuously in \( {\mathrm{H}}^{3} \) . | Proof. Suppose first that \( G \) is discrete. As \( G \) is the homomorphic image of a discrete (and therefore countable) subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \), we see that \( G \) is countable, say\n\n\[ G = \left\{ {{g}_{1},{g}_{2},\ldots }\right\} \]\n\nAs \( G \) is discrete, \( \begin{Vmatri... | Yes |
Lemma 5.3.3. Let \( G \) be any subgroup of \( \mathcal{M} \) and let \( D \) be an open subset of \( \widehat{\mathbb{C}} \) which contains a fixed point \( v \) of some parabolic or loxodromic element \( g \) of G. Then \( G \) does not act discontinuously in \( D \) . | Proof. This is trivial as the stabilizer \( {G}_{v} \) contains the distinct iterates of \( g \) . If \( g \) is parabolic or loxodromic, then \( {G}_{v} \) is infinite and this violates (5.3.5). | Yes |
Let \( G \) be Picard’s group, namely the group of transformations of the form\n\n\[ g\left( z\right) = \frac{{az} + b}{{cz} + d} \]\n\nwhere \( a, b, c \) and \( d \) are Gaussian integers (of the form \( m + {in} \) where \( m \) , \( n \in \mathbb{Z}) \) and \( {ad} - {bc} = 1 \). Obviously \( G \) is discrete.\n\nB... | Let \( w = \left( {p + {iq}}\right) /r \) where \( p, q \) and \( r \) are integers: obviously, the set of such \( w \) is dense in \( \widehat{\mathbb{C}} \). Now simply observe that\n\n\[ h\left( z\right) = \frac{\left( {1 - w{r}^{2}}\right) z + {r}^{2}{w}^{2}}{-{r}^{2}z + \left( {1 + w{r}^{2}}\right) } \]\n\n is a p... | Yes |
Lemma 5.3.5. Let \( \sum \) be an open disc and suppose that \( g \in \mathcal{M} \) and \( g\left( \bar{\sum }\right) \subset \sum \) . Then \( g \) is loxodromic and has a fixed point in \( g\left( \bar{\sum }\right) \) . | Proof. We may assume that \( g\left( \infty \right) = \infty \) . With this assumption, \( \partial \sum \) is a Euclidean circle (and not a straight line) as clearly, no fixed point of \( g \) is on the boundary of \( \sum \) . If \( g \) is elliptic or parabolic then (as \( g \) fixes \( \infty \) ) \( g \) is a Eucl... | Yes |
For any non-elementary group \( G \), the limit set \( \Lambda \) is the smallest non-empty \( G \) -invariant closed subset of \( \widehat{\mathbb{C}} \) . In addition, \( \Lambda \) is a perfect set and is therefore uncountable. | Proof. As \( {\Lambda }_{0} \) is \( G \) -invariant, so is \( \Lambda \) . By definition, \( \Lambda \) is closed and by Theorem 5.1.3, \( \Lambda \neq \varnothing \) . Now let \( E \) be any non-empty, closed \( G \) -invariant subset of \( \widehat{\mathbb{C}} \) . As \( G \) is non-elementary, every orbit is infini... | Yes |
Theorem 5.3.8. Let \( G \) be a non-elementary subgroup of \( \mathcal{M} \) and let \( {O}_{1} \) and \( {O}_{2} \) be disjoint open sets both meeting \( \Lambda \) . Then there is a loxodromic \( g \) in \( G \) with a fixed point in \( {O}_{1} \) and a fixed point in \( {O}_{2} \) . | Proof. Recall that if \( f \) is loxodromic with an attractive fixed point \( \alpha \) and a repulsive fixed point \( \beta \), then as \( n \rightarrow + \infty ,{f}^{n} \rightarrow \alpha \) uniformly on each compact subset of \( \widehat{\mathbb{C}} - \{ \beta \} \) and \( {f}^{-n} \rightarrow \beta \) uniformly on... | Yes |
Theorem 5.3.9. Let \( G \) be a non-elementary discrete subgroup of \( \mathcal{M} \). Then for all \( z \) in \( \widehat{\mathbb{C}} \), we have \( \Lambda = \Lambda \left( z\right) \). | Proof of Theorem 5.3.9. Each \( \Lambda \left( z\right) \) is closed, non-empty and \( G \) -invariant so by Theorem 5.3.7, we have\n\n\[ \Lambda \subset \Lambda \left( z\right) \]\n\nIf \( z \in \Lambda \), then \( G\left( z\right) \subset \Lambda \) and so\n\n\[ \Lambda \left( z\right) \subset \overline{G\left( z\rig... | Yes |
Theorem 5.3.10. Suppose that \( G \) is a discrete non-elementary subgroup of \( \mathcal{M} \). Then \( \Omega \) is the maximal domain of discontinuity in \( \widehat{\mathbb{C}} \) of \( G \): precisely,\n\n(i) \( G \) acts discontinuously in \( \Omega \); and\n\n(ii) if \( G \) acts discontinuously in an open subse... | Proof of Theorem 5.3.10. If \( G \) does not act discontinuously in \( \Omega \), then there is a compact subset \( K \) of \( \Omega \) and distinct \( {g}_{1},{g}_{2},\ldots \) in \( G \) such that \( {g}_{n}\left( K\right) \cap K \neq \varnothing \). Thus there are points \( {z}_{1},{z}_{2},\ldots \) in \( K \) with... | Yes |
Theorem 5.3.11. Suppose that \( G \) is non-elementary and that \( \Omega \neq \varnothing \) . If \( z \in \Omega \) then the stabilizer \( {G}_{z} \) is cyclic and finite. | Proof. By virtue of Lemma 5.3.3, if \( z \in \Omega \) then every element of the stabilizer \( {G}_{z} \) is either elliptic or \( I \) . Thus by Theorem 4.3.7, there is some \( \zeta \) in \( {H}^{3} \) which is fixed by every \( g \) in \( {G}_{z} \) . Now let \( A \) be the unique semi-circle in \( {H}^{3} \) which ... | Yes |
Theorem 5.3.12. Let \( G \) be a discrete non-elementary subgroup of \( \mathcal{M} \). Then (considering only \( g \) in \( G \) ):\n\n(i) each \( x \) in \( {H}^{3} \) is the centre of an open hyperbolic ball \( N \) such that \( g\left( N\right) = N \) if \( g\left( x\right) = x \) and \( g\left( N\right) \cap N = \... | Proof. First,(i) is a direct consequence of the fact that \( G \) is a group of isometries acting discontinuously in \( {\mathrm{H}}^{3} \).\n\nTo prove (ii), we may assume that \( z = 0 \) and that every \( g \) in \( {G}_{z} \) also fixes \( \infty \) (use Theorem 5.3.11). Now select a disc\n\n\[ N = \{ z : \left| z\... | Yes |
Theorem 5.3.14. Let \( G \) be a discrete non-elementary subgroup of \( \mathcal{M} \). (i) If \( D \) is a non-empty open \( G \) -invariant set which is not \( \widehat{\mathbb{C}} \), then \( G \) acts discontinuously in \( D \) ; (ii) if \( D \) is a non-empty open set such that \( g\left( D\right) \cap D = \varnot... | Proof. The set \( E = \widehat{\mathbb{C}} - D \) is non-empty, closed and \( G \) -invariant and so by Theorem 5.3.7, \( \Lambda \subset E \) . Thus \( G \) acts discontinuously in \( D \) (Theorem 5.3.10). By definition, \( \bigcup g\left( D\right) \) is disconnected and so is not \( \widehat{\mathbb{C}} \) : now app... | Yes |
Theorem 5.3.15. Let \( {G}_{1},{G}_{2},\ldots \) be subgroups of \( \mathcal{M} \) whose union generates the group \( G \) . Let \( {D}_{j} \) be a \( {G}_{j} \) -packing and suppose that \( {D}_{i} \cup {D}_{j} = \widehat{\mathbb{C}} \) when \( i \neq j \) . Suppose also that \( {D}^{ * }\left( { = \bigcap {D}_{j}}\ri... | Proof. Consider any element \( {g}_{n}\cdots {g}_{1} \) of \( G \) where \( {g}_{k} \in {G}_{{i}_{k}},{g}_{k} \neq I \) and \( {i}_{k} \neq {i}_{k + 1} \) for any \( k \) . First, because \( {D}_{{i}_{1}} \) is a \( {G}_{{i}_{1}} \) -packing, we have\n\n\[ \n{g}_{1}\left( {D}^{ * }\right) \subset {g}_{1}\left( {D}_{{i}... | Yes |
Theorem 5.4.1. (Jørgensen's Inequality). Suppose that the Möbius transformations \( f \) and \( g \) generate a discrete non-elementary group. Then\n\n\[ \n\\left| {{\\operatorname{tr}}^{2}\\left( f\\right) - 4}\\right| + \\left| {\\operatorname{tr}\\left( {{fg}{f}^{-1}{g}^{-1}}\\right) - 2}\\right| \\geq 1.\n\]\n\n(5.... | Proof of THEOREM 5.4.1. The idea of the proof is contained in Section 1.5 and Theorem 5.1.4. We know that \( \\langle f, g\\rangle \) is discrete and non-elementary. Now (5.4.1) holds if \( f \) is of order two (because then, \( {\\operatorname{tr}}^{2}\\left( f\\right) = 0 \) ) so we may assume that \( f \) is not of ... | No |
Theorem 5.4.2. A non-elementary group \( G \) of Möbius transformations is discrete if and only if for each \( f \) and \( g \) in \( G \), the group \( \langle f, g\rangle \) is discrete. | Proof. If \( G \) is discrete, then so is every subgroup of \( G \) . Now suppose that every subgroup \( \langle f, g\rangle \) is discrete: we suppose that \( G \) is not discrete and our aim is to reach a contradiction.\n\nAs \( G \) is not discrete we can find distinct \( {f}_{1},{f}_{2},\ldots \left( { \neq I}\righ... | Yes |
Theorem 5.4.3. Let \( f \) be parabolic and suppose that \( \langle f, g\rangle \) is discrete and non-elementary. Then\n\n(i)\n\n\[ \parallel f - I\parallel \cdot \parallel g - I\parallel \geq 1 \]\n\nand this is best possible;\n\n(ii) if \( g \) is also parabolic, then for all \( x \) in \( {H}^{3} \) we have\n\n\[ \... | Proof. There is a Möbius \( h \) corresponding to a unitary matrix \( U \) such that \( {hf}{h}^{-1} \) fixes \( \infty \) . If \( A \) corresponds to \( f \), then\n\n\[ \begin{Vmatrix}{{UA}{U}^{-1} - I}\end{Vmatrix} = \parallel A - I\parallel \]\n\nand similarly for \( g \) : thus we may assume that \( f \) fixes \( ... | Yes |
Theorem 5.4.5. Suppose that \( \langle g, h\rangle \) is discrete and non-elementary.\n\n(i) if \( g \) is parabolic, then for all \( x \) in \( {\mathrm{H}}^{3} \) ,\n\n\[ \sinh \frac{1}{2}\rho \left( {x,{gx}}\right) \sinh \frac{1}{2}\rho \left( {x,{\operatorname{hgh}}^{-1}x}\right) \geq \frac{1}{4}; \]\n\n(ii) if \( ... | If\n\n\[ \rho \left( {x,{gx}}\right) < \varepsilon ,\;\rho \left( {x,{hx}}\right) < \varepsilon ,\]\n\nthen\n\n\[ \rho \left( {x,{hg}{h}^{-1}x}\right) = \rho \left( {{h}^{-1}x, g{h}^{-1}x}\right) \]\n\n\[ \leq \rho \left( {{h}^{-1}x, x}\right) + \rho \left( {x,{gx}}\right) + \rho \left( {{gx}, g{h}^{-1}x}\right) \]\n\n... | No |
Theorem 6.3.3. Suppose that \( G \) acts discontinuously in \( D \) and that \( {D}_{0} \) is stable with stabilizer \( {G}_{0} \) . If either\n\n(i) \( {D}_{0} \) is open in \( D \) ; or\n\n(ii) \( {D}_{0}/{G}_{0} \) is compact;\n\nthen \( {D}_{0}/{G}_{0} \) (with the quotient topology) and \( \pi \left( {D}_{0}\right... | Proof. Both quotient maps\n\n\[ \pi : D \rightarrow D/G,\;\phi : {D}_{0} \rightarrow {D}_{0}/{G}_{0} \]\n\nare continuous and open as the respective groups are groups of homeomorphisms of the corresponding spaces. The restriction \( {\pi }_{0} \) of \( \pi \) to \( {D}_{0} \) is continuous so the natural bijection\n\n\... | Yes |
Suppose that \( G \) preserves and acts discontinuously in the upper half-plane \( {H}^{2} \) of \( \mathbb{C} \) and let \( g \) be a hyperbolic element of \( G \). We may assume that \( g \) fixes 0 and \( \infty \) so the positive imaginary axis, say \( L \), is invariant under \( g \). Suppose now that for all \( h... | Now \( g\left( z\right) = {kz} \), say, where \( k > 1 \), and \( L/\langle g\rangle \) is compact and, in fact, is a simple closed curve. According to Theorem 6.3.3, the projection of \( L \) into \( {H}^{2}/G \) is also a simple closed curve. | No |
Suppose that a group \( G \) acts discontinuously in a subdomain \( D \) of \( \widehat{\mathbb{R}} \) and that there is an open disc \( Q \) which is stable with stabilizer \( \langle g\rangle \) where \( g \) is parabolic. As \( Q \) is open, Theorem 6.3.3 implies that the projection of \( Q \) in \( D/G \) is confor... | By conjugation, we may assume that \( g\left( z\right) = z + 1 \) so that for some \( {y}_{0} \) ,\n\n\[ Q = \left\{ {x + {iy} : y > {y}_{0}}\right\} \]\n\nIt is clear that the quotient space \( Q/\langle g\rangle \) is conformally equivalent to the image of \( Q \) under the map \( z \mapsto \exp \left( {2\pi iz}\righ... | Yes |
Theorem 7.2.1. With \( \rho \) as above, and with \( z \), w in \( {H}^{2} \), (i) \[ \rho \left( {z, w}\right) = \log \frac{\left| {z - \bar{w}}\right| + \left| {z - w}\right| }{\left| {z - \bar{w}}\right| - \left| {z - w}\right| } \] (ii) \[ \cosh \rho \left( {z, w}\right) = 1 + \frac{{\left| z - w\right| }^{2}}{2\op... | Proof of Theorem 7.2.1. It is easy to see that the five equations are equivalent to each other: we shall prove that (ii) holds. By (7.2.2), the left-hand side of (ii) is invariant under \( g \) . A straightforward computation shows that \[ \frac{{\left| g\left( z\right) - g\left( w\right) \right| }^{2}}{\operatorname{I... | Yes |
Theorem 7.2.2. (i) The area of a hyperbolic disc of radius \( r \) is \( {4\pi }{\sinh }^{2}\left( {\frac{1}{2}r}\right) \) . (ii) The length of a hyperbolic circle of radius \( r \) is \( {2\pi }\sinh r \) . | Proof. We use the model \( \Delta \) and let \( C \) and \( D \) be the circle and disc with centre \( O \) and (hyperbolic) radius \( r \) . From (7.2.4) we see that\n\n\[ C = \{ z : \left| z\right| = R\} ,\;D = \{ z : \left| z\right| \leq R\} ,\]\n\nwhere\n\n\[ \sinh \left( {\frac{1}{2}r}\right) = \frac{R}{{\left( 1 ... | Yes |
Theorem 7.3.1. Let \( z \) and \( w \) be any points in the hyperbolic plane. A curve \( \gamma \) joining \( z \) to \( w \) satisfies\n\n\[ \parallel \gamma \parallel = \rho \left( {z, w}\right) \]\n\nif and only if \( \gamma \) is a parametrization of \( \left\lbrack {z, w}\right\rbrack \) as a simple curve. | It is for this reason that we refer to h-lines as geodesics (that is, curves of shortest length).\n\nNow consider any three points \( z, w \) and \( \zeta \) . It is clear from the special case (7.2.3) that if \( \zeta \) is between \( z \) and \( w \), then\n\n\[ \rho \left( {z, w}\right) = \rho \left( {z,\zeta }\righ... | No |
Theorem 7.7.2. Let \( L \) be the geodesic containing the longest side, say \( \left\lbrack {{z}_{2},{z}_{3}}\right\rbrack \) , of \( T \) . Then the geodesic \( {L}_{1} \) through \( {z}_{1} \) and orthogonal to \( L \) meets \( L \) at a point \( w \) in \( \left\lbrack {{z}_{2},{z}_{3}}\right\rbrack \) . | Proof. We may assume that \( L \) is the positive imaginary axis so \( w = i\left| {z}_{1}\right| \) : see Figure 7.7.2.\n\nIt is easy to see that\n\n\[ \rho \left( {{z}_{1},{z}_{2}}\right) \geq \rho \left( {w,{z}_{2}}\right) \]\n\nand similarly\n\n\[ \rho \left( {{z}_{1},{z}_{3}}\right) \geq \rho \left( {w,{z}_{3}}\ri... | Yes |
Theorem 7.9.1. Let \( T \) be a triangle with angles \( \alpha ,0,\pi /2\left( {\alpha \neq 0}\right) \) . Then\n\n(i) \( \sinh b\tan \alpha = 1 \) ;\n\n(ii) \( \cosh b\sin \alpha = 1 \) ;\n\n(iii) \( \tanh b\sec \alpha = 1 \) . | Proof. We work in \( {H}^{2} \) and we may assume that\n\n\[ \n{v}_{c} = i,\;{v}_{b} = \infty ,\;{v}_{a} = x + {iy},\n\]\n\nwhere \( {x}^{2} + {y}^{2} = 1 \) : see Figure 7.9.1. As \( y = \sin \alpha \), Theorem 7.2.1(ii) yields (ii). The remaining formulae are equivalent to (ii). | No |
Theorem 7.10.1. For any triangle with angles \( \alpha ,\beta ,0 \) we have\n\n(i)\n\n\[ \cosh c = \frac{1 + \cos \alpha \cos \beta }{\sin \alpha \sin \beta } \]\n\n(ii)\n\n\[ \sinh c = \frac{\cos \alpha + \cos \beta }{\sin \alpha \sin \beta }. \] | Proof. We work in \( {H}^{2} \) with \( {v}_{c} = \infty \) . We may assume that \( {v}_{a} \) and \( {v}_{b} \) lie on the circle \( \left| z\right| = 1 \), say with\n\n\[ {v}_{a} = \exp \left( {i\theta }\right) ,\;{v}_{b} = \exp \left( {i\phi }\right) \]\n\nwhere \( 0 < \theta < \phi < \pi \) . Thus \( \alpha = \thet... | No |
Theorem 7.11.1. For any triangle with angles \( \alpha ,\beta ,\pi /2 \) we have\n\n\[ \cosh c = \cosh a\cosh b. \] | Proof. Using Theorem 7.2.1(ii) we have\n\n\[ \cosh c = \left( {1 + {k}^{2}}\right) /{2kt} \]\n\n\[ \cosh b = 1/t \]\n\n\[ \cosh a = \left( {1 + {k}^{2}}\right) /{2k} \] | Yes |
Theorem 7.11.2. For any triangle with angles \( \alpha ,\beta ,\pi /2 \) we have\n\n(i) \( \tanh b = \sinh a\tan \beta \) ;\n\n(ii) \( \sinh b = \sinh c\sin \beta \) ;\n\n(iii) \( \tanh a = \tanh c\cos \beta \) . | Proof. Let the geodesic through \( {v}_{a} \) and \( {v}_{b} \) have Euclidean centre \( {x}_{0} \) . Then by equating the distances from \( {v}_{a} \) and \( {v}_{b} \) to \( {x}_{0} \) we see that\n\n\[ {k}^{2} = 1 - 2{x}_{0}s \]\n\nThis shows that \( {x}_{0} < 0 \) . The Euclidean triangle with vertices \( {x}_{0},0... | No |
Theorem 7.11.3. For any triangle with angles \( \alpha ,\beta ,\pi /2 \) :\n\n(i)\n\n\[ \cosh a\sin \beta = \cos \alpha \]\n\n(ii)\n\n\[ \cosh c = \cot \alpha \cot \beta \text{.} \] | Proof. Theorem 7.11.2(i) gives\n\n\[ \sinh a\tan \beta = \tanh b, \]\n\n\[ \sinh b\tan \alpha = \tanh a \]\n\nand elimination of \( b \) gives (i).\n\nTo prove (ii), simply eliminate cosh \( a \) and cosh \( b \) from (7.11.1), Theorem 7.11.3(i) and the corresponding identity with \( a \) and \( \alpha \) interchanged ... | No |
Theorem 7.13.1. For any triangle \( T \) with angles \( \alpha ,\beta \) and \( \gamma \) , \n\n\[ \n\text{h-area}\left( T\right) = \pi - \left( {\alpha + \beta + \gamma }\right) \text{.} \n\] | Proof. Assume first that \( \gamma = 0 \) . We may assume that \( {v}_{c} = \infty \) and that \( {v}_{a} \) and \( {v}_{b} \) lie on \( \left| z\right| = 1 \) . Referring to Figure 7.13.1 we find that \n\n\[ \n\text{h-area}\left( T\right) = {\int }_{\cos \left( {\pi - \alpha }\right) }^{\cos \beta }\left\lbrack {{\int... | Yes |
Theorem 7.14.1. The three angle bisectors of a triangle \( T \) meet at a point \( \zeta \) in \( T \) . | Proof. We may assume that \( \gamma \) is the smallest angle so \( \gamma < \pi /2 \) . Now construct angle bisectors at \( {v}_{a} \) and \( {v}_{b} \) : these must meet at a point \( \zeta \) in \( T \) (see Section 7.7). Next, define \( {\gamma }_{1} \) and \( {\gamma }_{2} \) as in Figure 7.14.1. As \( \alpha /2,\b... | Yes |
Theorem 7.14.2. The radius \( R \) of the inscribed circle of \( T \) is given by\n\n\[ \n{\tanh }^{2}R = \frac{{\cos }^{2}\alpha + {\cos }^{2}\beta + {\cos }^{2}\gamma + 2\cos \alpha \cos \beta \cos \gamma - 1}{2\left( {1 + \cos \alpha }\right) \left( {1 + \cos \beta }\right) \left( {1 + \cos \gamma }\right) }.\n\] | Proof. Let \( x = \rho \left( {{v}_{a},{w}_{c}}\right) \) and \( y = \rho \left( {{w}_{c},{v}_{b}}\right) \) . Then\n\n\[ \n\frac{\cos \alpha \cos \beta + \cos \gamma }{\sin \alpha \sin \beta } = \cosh x\cosh y + \sinh x\sinh y\n\]\n\nso\n\n\[ \n{\left\lbrack \left( \cos \alpha \cos \beta + \cos \gamma \right) - \left(... | Yes |
For each \( \alpha \) in \( \left( {0,\pi }\right) \) we can construct a triangle \( T \) with angles \( \alpha ,0,0 \) . Then\n\n\[ 4{\tanh }^{2}R = \frac{1}{2}\left( {1 + \cos \alpha }\right) \] | \[ 4{\tanh }^{2}R = \frac{1}{2}\left( {1 + \cos \alpha }\right) \]\n\n\[ = {\cos }^{2}\left( {\alpha /2}\right) \]\n\n\[ = {\sin }^{2}\left\lbrack {\frac{1}{2}\mathrm{\;h} - \operatorname{area}\left( T\right) }\right\rbrack \text{.} \] | Yes |
Theorem 7.14.4. The radius \( R \) of the inscribed circle of \( T \) satisfies\n\n\[ \n\tanh R \geq \frac{1}{2}\sin \left\lbrack {\frac{1}{2}\mathrm{h} - \mathrm{{area}}\left( T\right) }\right\rbrack \n\]\n\nand this lower bound is best possible for each value of \( h \) -area(T). | Example 7.14.3 shows that this lower bound is best possible. | No |
Theorem 7.15.1. If \( P \) is any polygon with interior angles \( {\theta }_{1},\ldots ,{\theta }_{n} \), then\n\n\[ \n\text{h-area}\left( P\right) = \left( {n - 2}\right) \pi - \left( {{\theta }_{1} + \cdots + {\theta }_{n}}\right) \text{.} \n\] | Proof. This has been proved for the case \( n = 3 \) (Section 7.13) and from this it follows for convex polygons by subdivision of \( P \) into \( n - 2 \) triangles (the details are omitted). It is worth noting explicitly that Theorem 15.1 applies to all polygons whether convex or not.\n\nThe proof for non-convex poly... | Yes |
Theorem 7.16.1. Let \( P \) be a polygon with interior angles \( {\theta }_{1},\ldots ,{\theta }_{n} \) . Then \( P \) is convex if and only if each \( {\theta }_{j} \) satisfies \( 0 \leq {\theta }_{j} \leq \pi \) . | This is an immediate consequence of Theorem 7.5.1. | No |
Theorem 7.16.2. Let \( {\theta }_{1},\ldots ,{\theta }_{n} \) be any ordered \( n \) -tuple with \( 0 \leq {\theta }_{j} < \pi \) , \( j = 1,\ldots, n \) . Then there exists a polygon \( P \) with interior angles \( {\theta }_{1},\ldots ,{\theta }_{n} \) , occurring in this order around \( \partial P \), if and only if... | Proof. Given \( {\theta }_{1},\ldots ,{\theta }_{n} \) satisfying (7.16.1) and each lying in \( \lbrack 0,\pi ) \), construct quadrilaterals \( {Q}_{1},\ldots ,{Q}_{n} \) each with one vertex at the origin in \( \Delta \) as in Figure 7.16.1. The length \( d \) can take any positive value and is to be determined later:... | Yes |
Theorem 7.17.1. (i) \( \sinh {a}_{1}\sinh {a}_{2} = \cos \phi \) ; (ii) \( \cosh {a}_{1} = \cosh {b}_{1}\sin \phi \) . | The proof depends on two useful preliminary results. | No |
Lemma 7.17.2. Let \( L \) be a hyperbolic geodesic in \( \Delta \) with Euclidean centre \( \xi \) and radius \( r \) and let \( w \) be the point on \( L \) which is nearest to the origin. Then\n\n\[ \sinh \rho \left( {0, w}\right) = 1/r,\;\cosh \rho \left( {0, w}\right) = \left| \xi \right| /r. \] | Proof. Clearly, \( \left| \xi \right| = \left| w\right| + r \) and orthogonality gives \( {\left| \xi \right| }^{2} = 1 + {r}^{2} \) . Using (7.2.4) we obtain \( \sinh \frac{1}{2}\rho \left( {0, w}\right) \) and hence\n\n\[ \sinh \rho \left( {0, w}\right) = \frac{2\left| w\right| }{1 - {\left| w\right| }^{2}} = \frac{1... | Yes |
Lemma 7.17.3. Let \( L \) and \( {L}^{\prime } \) be geodesics in the hyperbolic plane. Then the inversive product \( \left( {L,{L}^{\prime }}\right) \) is\n\n\[ \cosh \rho \left( {L,{L}^{\prime }}\right) ,\;1,\;\cos \phi \] \n\naccording as \( L \) and \( {L}^{\prime } \) are disjoint, parallel or intersecting at an a... | Proof. It is not difficult to see that disjoint geodesics have a common orthogonal geodesic (see Section 7.22) and (for the moment) \( \rho \left( {L,{L}^{\prime }}\right) \) is defined to be the length of this orthogonal segment between \( L \) and \( {L}^{\prime } \) . By the usual invariance arguments we need only c... | Yes |
Theorem 7.18.1. (i) \( \cosh a\cosh c + \cos \phi = \sinh a\cosh b\sinh c \) . | Proof. It is easy to see that there is a geodesic through the vertex with angle \( \phi \) which meets and is orthogonal to the side of length \( b \) . Let \( {b}_{1} \) and \( {b}_{2} \) be the lengths as illustrated and let \( {\phi }_{1},{\phi }_{2} \) be the subdivision of \( \phi ;{\phi }_{1} \) being on the same... | Yes |
Theorem 7.19.1.\n\n\[ \n\frac{\sinh {a}_{1}}{\sinh {b}_{1}} = \frac{\sinh {a}_{2}}{\sinh {b}_{2}} = \frac{\sinh {a}_{3}}{\sinh {b}_{3}}. \n\] | Proof. From Theorem 7.18.1 we obtain\n\n\[ \n\sinh {b}_{2}\sinh {a}_{3} = \cosh t = \sinh {a}_{2}\sinh {b}_{3} \n\]\n\nand the result follows by symmetry considerations. | No |
Theorem 7.19.2.\n\n\( \cosh {b}_{1}\sinh {a}_{2}\sinh {a}_{3} = \cosh {a}_{1} + \cosh {a}_{2}\cosh {a}_{3} \) | Proof. From Theorem 7.18.1 we obtain the identities\n\n\[ \sinh x\sinh {a}_{2} = \cosh u \]\n\n\[ \sinh y\sinh {a}_{3} = \cosh v \]\n\n\[ \sinh u\sinh t = \cosh {a}_{2} \]\n\n\( \sinh v\sinh t = \cosh {a}_{3}. \)\n\nNext, we obtain the identity\n\n\[ \left( {{\cosh }^{2}{a}_{2} + {\sinh }^{2}u}\right) \left( {{\cosh }^... | Yes |
Theorem 7.35.1. (i) If \( g \) is hyperbolic with axis \( A \) and translation length \( T \) then\n\n\[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) = \cosh \rho \left( {z, A}\right) \sinh \left( {\frac{1}{2}T}\right) . \] | Proof. By conjugation, we may assume in (i) that \( g \) acts on \( {H}^{2} \) and that \( g\left( z\right) = {kz}, k > 1 \) . By Theorem 7.2.1 we have\n\n\[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) = \frac{\left| z - kz\right| }{{2y}\sqrt{k}} \]\n\n\[ = \left( \frac{\left| z\right| }{y}\right) \frac{1}{2}\left( {\... | Yes |
Theorem 7.37.2. (i) \( {\sum }_{g} \) is conjugation invariant: explicitly\n\n\[ \n{\sum }_{{hg}{h}^{-1}} = h\left( {\sum }_{g}\right) \n\]\n\n(ii) \( {\sum }_{g} \) determines the pair \( \left\{ {g,{g}^{-1}}\right\} \) : explicitly, \( {\sum }_{g} = {\sum }_{h} \) if and only if \( h = g \) or \( h = {g}^{-1} \) . | Proof of THEOREM 7.37.2. First, (i) is trivially true. Next, observe from the geometric construction of \( {\sum }_{g} \) that \( {\sum }_{g} \) determines the fixed points of \( g \) and also the pairs \( \{ z,{gz}\} \) on the circle at infinity. It follows that \( {\sum }_{g} \) determines the pair \( \left\{ {g,{g}^... | No |
Theorem 7.38.1. Let \( {L}_{1} \) and \( {L}_{2} \) be distinct geodesics, let \( {\sigma }_{j} \) denote reflection in \( {L}_{j} \) and let \( f = {\sigma }_{1}{\sigma }_{2} \) . Then the inversive product \( \left( {{L}_{1},{L}_{2}}\right) \) satisfies\n\n\[ \left( {{L}_{1},{L}_{2}}\right) = \frac{1}{2}\left| {\oper... | Proof. If \( {L}_{1} \) and \( {L}_{2} \) are disjoint, then their common orthogonal geodesic \( L \) is invariant under \( {\sigma }_{1} \) and \( {\sigma }_{2} \) . It follows that \( f \) is hyperbolic, that \( L \) is the axis of \( f \) and consequently, the translation length \( T \) of \( f \) satisfies\n\n\[ \f... | Yes |
Theorem 7.38.2. Let \( g \) and \( h \) be elliptic isometries with \( g \) a rotation of \( {2\theta } \) about \( u \) and \( h \) a rotation of \( {2\phi } \) about \( v \) . We suppose that \( g \) and \( h \) are rotations in the same sense with \( u \neq v \) and \( \theta ,\phi \) in \( \left( {0,\pi }\right) \)... | Proof. We may assume that \( g \) and \( h \) act on \( {H}^{2} \), that \( u \) and \( v \) lie on the positive imaginary axis \( L \) and that\n\n\[ \ng = {\sigma }_{1}{\sigma }_{2},\;h = {\sigma }_{3}{\sigma }_{4},\n\]\n\nwhere \( {L}_{2} = L = {L}_{3} \) . This is illustrated in Figure 7.38.1 and by Theorem 7.38.1,... | Yes |
Corollary 7.38.5. Suppose that \( g \) and \( h \) are hyperbolic with disjoint axes and the same translation length \( T \) . If \( {gh} \) and \( g{h}^{-1} \) are not elliptic, then\n\n\[ \sinh \frac{1}{2}\rho \left( {{A}_{g},{A}_{h}}\right) \sinh \left( {\frac{1}{2}T}\right) \geq 1. \] | Proof. With these assumptions we have\n\n\[ \frac{1}{2}\left| {\operatorname{trace}\left( {gh}\right) }\right| \geq 1 \]\n\nand similarly for \( g{h}^{-1} \) . By using \( h \) or \( {h}^{-1} \) we may assume that \( \varepsilon = - 1 \) in Theorem 7.38.3 and the result follows as\n\n\[ \cosh \rho \left( {{A}_{g},{A}_{... | Yes |
Theorem 7.38.6. Let \( g \) and \( h \) be hyperbolic and suppose that \( {A}_{g} \) and \( {A}_{h} \) intersect at a point \( {v}_{2} \) in an angle \( \theta ,0 < \theta < \pi \), this being the angle between the half-rays from \( {v}_{2} \) to the attractive fixed points of \( g \) and \( h \) . Then \( {gh} \) is h... | Proof. This proof uses the alternative expression of a hyperbolic element as a product of two rotations of order two (see Section 7.34).\n\n\n\nFigure 7.38.4\n\nWe refer to Figure 7.38.4: then\n\n\[ \n{gh} = \left( {... | Yes |
Theorem 7.39.1. Let \( g \) be parabolic and suppose that \( g \) and \( h \) have no common fixed point. Then \( \left\lbrack {g, h}\right\rbrack \) is hyperbolic. | Proof. A matrix proof (with \( g\left( z\right) = z + 1 \) ) is easy enough but the geometry is more revealing. Let \( g \) fix the point \( v \) and let \( {L}_{2} \) be the geodesic from \( v \) to \( h\left( v\right) \) . For a suitable \( {L}_{1} \) and \( {L}_{3} \) we can write\n\n\[ g = {\sigma }_{1}{\sigma }_{2... | Yes |
Theorem 7.39.2. Let \( g \) be elliptic with fixed point \( v \) and angle of rotation \( {2\theta } \) , \( 0 < \theta \leq \pi \) . Let \( h \) be any isometry not fixing \( v \) : then \( \left\lbrack {g, h}\right\rbrack \) is hyperbolic with translation length \( T \) and\n\n\[ \sinh \left( {T/4}\right) = \sinh \fr... | Proof. We write \( g = {\sigma }_{1}{\sigma }_{2} \) where \( {L}_{2} \) joins \( v \) to \( h\left( v\right) \) . Now construct \( {L}_{3} \) as in Figure 7.39.1 so \( h{g}^{-1}{h}^{-1} = {\sigma }_{2}{\sigma }_{3} \) and \( \left\lbrack {g, h}\right\rbrack = {\sigma }_{1}{\sigma }_{3} \).\n\n![32ff4eba-fdcc-4eb0-a03c... | Yes |
Theorem 7.39.3. Let \( g \) and \( h \) be hyperbolic and suppose that \( h\left( {A}_{g}\right) \) and \( {A}_{g} \) cross at an angle \( \theta \) (between the positive directions of \( g \) and \( h{g}^{-1}{h}^{-1} \) ). Then \( \left\lbrack {g, h}\right\rbrack \) is hyperbolic with translation length \( T \) where\... | Proof. Apply Theorem 7.38.6 with \( h \) in that theorem replaced by \( h{g}^{-1}{h}^{-1} \) : thus\n\n\[ \cosh \left( {\frac{1}{2}T}\right) = {\cosh }^{2}\left( {\frac{1}{2}{T}_{g}}\right) + {\sinh }^{2}\left( {\frac{1}{2}{T}_{g}}\right) \cos \theta . \] | Yes |
Theorem 7.39.4. Let \( g \) and \( h \) be hyperbolic with their axes \( {A}_{g} \) and \( {A}_{h} \) crossing at an angle \( \theta ,0 < \theta < \pi \) . If \( \left\lbrack {g, h}\right\rbrack \) is not elliptic then\n\n\[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \sin \theta \... | Proof. The situation is that described in one of the last two diagrams in Figure 7.39.2. We may apply Theorem 7.38.3 with \( h \) in that theorem replaced\n\n\n\nFigure 7.39.2\n\n\n\nby \( h{g}^{-1}{h}^{-1} \) and wi... | Yes |
Corollary 7.39.5. Let \( {g}_{1},\ldots ,{g}_{n} \) be conjugate hyperbolic elements in a group \( G \) with no elliptic elements, let \( T \) be the common translation length and suppose that the axes \( {A}_{j} \) of \( {g}_{j} \) are concurrent. Then\n\n\[ \n{\sinh }^{2}\left( {\frac{1}{2}T}\right) \sin \left( {\pi ... | Proof. Two axes \( {A}_{i} \) and \( {A}_{j} \) must cross at an angle \( \theta \) where \( \theta \leq \pi /n \) : now apply Theorem 7.39.4. | No |
Theorem 8.2.1. Let \( G \) be a purely hyperbolic group with \( \Delta \) as its invariant disc. Then \( G \) is either discrete or elementary. Further, if \( g, h \in G \) and \( \langle g, h\rangle \) is non-elementary, then for all \( z \) in \( \Delta \) , | \[ \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) \sinh \frac{1}{2}\rho \left( {z,{hz}}\right) \geq 1. \] (8.2.1) The lower bound is best possible. | No |
For \( z = 0 \), the inequality in Theorem 8.2.1 is\n\n\[ \n{\left| g\left( 0\right) \right| }^{2} \cdot {\left| h\left( 0\right) \right| }^{2} \geq \left( {1 - {\left| g\left( 0\right) \right| }^{2}}\right) \left( {1 - {\left| h\left( 0\right) \right| }^{2}}\right)\n\] | and this is equivalent to the next inequality (which is a Euclidean version of Theorem 8.2.1). | No |
Example 8.2.5. Construct four disjoint geodesics \( {L}_{j} \) in \( \Delta \) as in Figure 8.2.1. Let \( g \) be the hyperbolic element which fixes \( 1, - 1 \) and which maps \( {L}_{1} \) to \( {L}_{2} \) : let \( h \) be the hyperbolic element which fixes \( i, - i \) and which maps \( {L}_{3} \) to \( {L}_{4} \) a... | Using Corollary 5.3.15 (with \( {G}_{1} = \langle g\rangle ,{G}_{2} = \langle h\rangle \) and \( D \) the region bounded by the \( {L}_{j} \) ), we see that \( G \) acts discontinuously in \( \Delta \) . It will be apparent from later considerations (Chapter 9) that \( G \) is purely hyperbolic ( \( D \) is a fundament... | No |
Theorem 8.4.1. Let \( G \) be a non-elementary group of isometries of the hyperbolic plane: the following statements are equivalent.\n\n(1) \( G \) is discrete;\n\n(2) \( G \) acts discontinuously in \( \Delta \) ;\n\n(3) the fixed points of elliptic elements of \( G \) do not accumulate in \( \Delta \) ;\n\n(4) the el... | Proof that (2) IMPLIES (3). Select any \( z \) in \( \Delta \) and any compact neighbourhood \( N \) of \( z \) . By (2), \( g\left( N\right) \) meets \( N \) for only a finite set of \( g \) in \( G \) so only finitely many fixed points lie in \( N \) .\n\nProof that (3) implies (5). If (5) fails, then \( G \) contain... | No |
Theorem 8.5.2. \( N \) is the smallest non-empty \( G \) -invariant open convex subset of \( \Delta \) . | Proof. As \( N \) has these properties except possibly of being the smallest such set, we must show that any non-empty \( G \) -invariant open convex set \( E \) contains \( N \) . As \( E \) is non-empty and \( G \) -invariant, it contains some \( G \) -orbit which necessarily accumulates at each point of \( \Lambda \... | Yes |
Theorem 9.1.3. Let \( {F}_{1} \) and \( {F}_{2} \) be measurable fundamental sets for \( G \) . Then\n\n\[ \n\mathrm{h} - \operatorname{area}\left( {F}_{1}\right) = \mathrm{h} - \operatorname{area}\left( {F}_{2}\right) .\n\]\n\nLet \( {F}_{0} \) be a measurable fundamental set for a subgroup \( {G}_{0} \) of index \( k... | Proof. Denote h-area by \( \mu \) . As \( \mu \) is invariant under each isometry we have\n\n\[ \n\mu \left( {F}_{1}\right) = \mu \left( {{F}_{1} \cap \left\lbrack {\mathop{\bigcup }\limits_{g}g{F}_{2}}\right\rbrack }\right)\n\]\n\n\[ \n= \mathop{\sum }\limits_{g}\mu \left( {{F}_{1} \cap g{F}_{2}}\right)\n\]\n\n\[ \n= ... | Yes |
Proposition 9.2.2. (i) \( \theta \) and \( \tau \) are injective;\n\n(ii) \( \pi ,\widetilde{\pi } \) and \( \theta \) are surjective;\n\n(iii) \( \pi ,\widetilde{\pi },\theta \) and \( \tau \) are continuous;\n\n(iv) \( \pi \) is an open map. | Proof. The only assertion which is not completely trivial is that \( \theta \) is continuous. If \( A \) is any open subset of \( \Delta /G \), we may apply (9.2.1) to obtain\n\n\[{\widetilde{\pi }}^{-1}\left( {{\theta }^{-1}A}\right) = \widetilde{D} \cap {\pi }^{-1}\left( A\right)\]\n\nand this is open in \( \widetild... | No |
We shall exhibit a convex five-sided polygon which is a fundamental domain for a Fuchsian group \( G \) but which is not locally finite. The group \( G \) is the group acting on \( {H}^{2} \) and generated by \[ f\left( z\right) = {2z},\;g\left( z\right) = \frac{{3z} + 4}{{2z} + 3}. \] | Our first task is to show that \( G \) is discrete and to identify a fundamental domain for \( G \) . To do this, consider Figure 9.2.3. A computation shows that \( f\left( {\gamma }_{1}\right) = {\gamma }_{2} \) and \( g\left( {\sigma }_{1}\right) = {\sigma }_{2} \) and a straightforward application of Theorem 5.3.15 ... | No |
Theorem 9.2.6. Let \( D \) be a fundamental domain for a Fuchsian group \( G \) and suppose that for each \( z \) in \( \partial D \) we have\n\n(1) there is some \( g \) in \( G \) with \( g \neq I \) and \( g\left( z\right) \in \partial D \) ;\n\n(2) \( z \) can be joined to a point in \( D \) by a curve lying entire... | Proof. Neither (1) nor (2) is sufficient to ensure that \( D \) is locally finite. We shall restrict ourselves here to a brief sketch of the proof in the most interesting case, namely when \( D \) is convex (convexity being stronger than (2)).\n\nIt is convenient to say that \( z \) in \( \Delta \) is regular if there ... | Yes |
Theorem 9.2.7. Let \( D \) be any locally finite fundamental domain for a Fuchsian group \( G \) . Then\n\n\[ \n{G}_{0} = \{ g \in G : g\left( \widetilde{D}\right) \cap \widetilde{D} \neq \varnothing \} \]\n\ngenerates \( G \) . | Proof. Let \( {G}^{ * } \) be the group generated by \( {G}_{0} \) . We may suppose that \( G \) acts in \( \Delta \) so for any \( z \) in \( \Delta \) there is some \( g \) in \( G \) with \( g\left( z\right) \in \widetilde{D} \) . Suppose also that \( h\left( z\right) \in \widetilde{D} \) . Then \( h\left( z\right) ... | Yes |
Theorem 9.2.8. Let \( D \) by any locally finite fundamental domain for a Fuchsian group \( G \). (i) Let \( g \) be an elliptic element in \( G \) and let \( K \) be a compact disc with \( g\left( K\right) = K \). Then \( \widetilde{D} \) meets a positive but only finite number of distinct images of \( K \). (ii) Let ... | Proof. In all cases, choose \( w \) in \( K \). For some \( h \) in \( G, h\left( w\right) \in \widetilde{D} \) so \( \widetilde{D} \) meets some image of \( K \). Now (i) is trivial for \( K \) is compact since if \( \widetilde{D} \) meets \( h\left( K\right) \), then \( {h}^{-1}\left( \widetilde{D}\right) \) meets \(... | Yes |
Corollary 9.2.9. Let \( G \) be a Fuchsian group, D any locally finite fundamental domain for \( G \) and let \( \zeta \) be fixed by some parabolic element of \( G \) . Then for some \( g \) in \( G, g\left( \zeta \right) \) lies in the Euclidean closure of \( D \) . | Proof. We may suppose that \( G \) acts on \( {H}^{2} \), that \( \zeta = \infty \) and that the stabilizer of \( \zeta \) is generated by \( p : z \mapsto z + 1 \) .\n\nNow let \( K \) be a horocyclic region invariant under \( p \) . Choose any sequence of points \( {z}_{1},{z}_{2},\ldots \) in \( K \) with \( \operat... | Yes |
Theorem 9.3.3. The side-pairing elements \( {G}^{ * } \) of \( P \) generate \( G \) . | Proof. Because of Theorem 9.2.7, it is only necessary to show that if \( \widetilde{P} \cap h\left( \widetilde{P}\right) \neq \varnothing \), then \( h \) lies in the group generated by the \( {g}_{s} \) . Consider, then, any \( w \) in \( \widetilde{P} \cap h\left( \widetilde{P}\right) \) . First, there is an open dis... | Yes |
For every Fuchsian group \( G \), every convex fundamental polygon \( P \) and every cycle \( C \), \[ \theta \left( C\right) = {2\pi }/\operatorname{ord}\left( C\right) \] | Proof. Without doubt, the most efficient description of the proof is by means of cosets. Let \( C = \left\{ {{z}_{1},\ldots ,{z}_{n}}\right\} \) so that for some \( {g}_{1}\left( { = I}\right) ,{g}_{2},\ldots ,{g}_{k} \) we have \( {g}_{j}\left( {z}_{j}\right) = {z}_{1} \) . It follows that \( {g}_{j}\left( P\right) \)... | Yes |
Theorem 9.3.8. Let \( v \) be any point of \( E \) that is fixed by some non-trivial element of \( G \) . Then \( v \) is fixed by a parabolic element of \( G \) (and not by any hyperbolic element). Further, the cycle \( C \) of \( v \) on \( E \) is a finite cycle each point of which is a proper vertex of \( P \) . | Proof. First, \( v \) cannot be fixed by an elliptic element in \( G \) as \( \left| v\right| = 1 \) . If \( v \) is fixed by a hyperbolic element \( h \) of \( G \), let \( A \) be the axis of \( h \) and construct any \( \lbrack z, v) \) in \( P \) . Take \( {z}_{n} \) on \( \lbrack z, v) \) with \( {z}_{n} \rightarr... | Yes |
Theorem 9.4.3. Let \( \\left\\{ {{z}_{1},\\ldots ,{z}_{n}}\\right\\} \) be any cycle on the boundary of the Dirichlet polygon \( D\\left( w\\right) \) . Then\n\n\[ \n\\rho \\left( {{z}_{1}, w}\\right) = \\rho \\left( {{z}_{2}, w}\\right) = \\cdots = \\rho \\left( {{z}_{n}, w}\\right) .\n\] | Proof. Consider, for example \( {z}_{1} \) and \( {z}_{2} \) on the boundary of \( D\\left( w\\right) \) with \( {z}_{2} = h\\left( {z}_{1}\\right) \) . As \( \\left\\lbrack {w,{z}_{1}}\\right) \\subset D\\left( w\\right) \) we see that\n\n\[ \n\\left\\lbrack {{hw},{z}_{2}}\\right) = h\\left\\lbrack {w,{z}_{1}}\\right)... | Yes |
Let \( G \) be the Modular group acting in \( {H}^{2} \) : we shall show that the open polygon \( P \) illustrated in Figure 9.4.2 is the Dirichlet polygon with centre \( {iv} \) for any \( v > 1 \) . | First, the isometries\n\n\[ f\left( z\right) = z + 1,\;g\left( z\right) = - 1/z \]\n\nare in \( G \) and (as the reader can easily verify) the three geodesic sides of \( P \) are \( {L}_{f},{L}_{{f}^{-1}},{L}_{g} \) . This shows that \( D \subset P \) .\n\nIf \( D \neq P \), then some side of \( D \) crosses \( P \) an... | Yes |
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