Geometry, topology and dynamics of character varieties by William Goldman, Caroline Series, Ser Peow Tan

By William Goldman, Caroline Series, Ser Peow Tan

This quantity is predicated on lectures given on the hugely winning three-week summer time institution on Geometry, Topology and Dynamics of personality forms held on the nationwide collage of Singapore's Institute for Mathematical Sciences in July 2010.

geared toward graduate scholars within the early phases of study, the edited and refereed articles include a superb advent to the topic of this system, a lot of that's in a different way on hand in basic terms in really good texts. themes contain hyperbolic constructions on surfaces and their degenerations, purposes of ping-pong lemmas in a number of contexts, introductions to Lorenzian and intricate hyperbolic geometry, and illustration types of floor teams into PSL(2, R) and different semi-simple Lie teams. This quantity will function an invaluable portal to scholars and researchers in a colourful and multi-faceted region of arithmetic.

Readership: Graduate scholars, researchers and professors in mathematical components similar to low-dimensional topology, dynamical platforms and hyperbolic geometry

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Extra resources for Geometry, topology and dynamics of character varieties

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L l2 l3 Fig. 20. P l P Q 1 .. .... .. ... . 1 . . . . . . . . . . . . . . ..... 1 .... .... . .. 2.................................................................. .......... 3 .. .. ... .. .. 8. Isometries as projective transformations Viewed in projective geometry, an isometry φ of H 2 extends naturally to a projective transformation Φ of the projective plane P 2 associated to H 2 , namely, a bijective self-mapping Φ of P 2 that sends any projective line to a projective line.

If the two straight lines are perpendicular, then the conclusion is obviously true. Otherwise, suppose the two straight lines form an acute angle P OQ with P Q ⊥ OQ. 9, as P moves in ray OP away from O, the distance |P Q| increases. Now take a point P1 on ray OP and produce OP1 to a sequence of points P2 , · · · , Pn , · · · on ray OP so that |OP1 | = |P1 P2 | = |P2 P3 | = · · · = |Pn−1 Pn | = · · · . For each n = 1, 2, · · · , draw Pn Qn ⊥ OQ with foot Qn on OQ. 7, we have |P2 Q2 | > 2|P1 Q1 |, and inductively, |P2n Q2n | > 2n |P1 Q1 | for n = 1, 2, · · · .

41. In a hyperbolic plane an equidistant curve is not a straight line. Hint: The proof is similar to that for a horocycle or a circle. 42. A generalized circle in a hyperbolic plane is either a circle, a horocycle, or an equidistant curve. The pencil consisting of all the straight lines perpendicular to a generalized circle is called its radii pencil. 43 (Circumcircle Theorem). Through the three vertices of a triangle in a hyperbolic plane passes a unique generalized circle, called the generalized circumcircle of the triangle.

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