MATH 113 FINAL EXAM
NAME:
1.(5 points) Among all the things that you have learned in the course, what impresses you the most?
Why?
2.(5 points) Calculate the hyperbolic distance between i and

2
2 +

2
2 i.
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1
23.(8 points) Let Γ ≤ Isom(R2) be a subgroup of type IIIb with generators (F, ae1/2) and (I, be2) (see
Proposition 3.3.5).
(1) Show there is a natural one-to-one correspondence between the quotient space R2/Γ and the half-
open-half-closed rectangle [0, a/2)× [0, b).
(2) Describe how the portals are built on the boundary of [0, a/2]× [0, b].
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34.(6 points) Find all isometries of (H, d) that preserves the pure imaginary half line L.
Remark: g “preserves” L means g · L = L. You can directly cite Lemma 5.2.1, which covers half of all
the isometries.
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45.(14 points) Let g be an isometry of the second kind that has no fixed points.
(1) Show that g must preserve a hyperbolic line.
(2) Show that g is conjugate to z 7→ −az¯, where a ∈ (0, 1) ∪ (1,∞).
Hint 1: To find the desired hyperbolic line, you may regard g as a map with domain D = H ∪ R.
Hint 2: For (2), make use of your result in the last problem.
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56.(12 points) Let Γ be a subgroup of Isom(R2) of type IIIa. Recall that this means that Γ can be
generated by two linearly independent translations (I, v1) and (I, v2). For R > 0, we define
Γ(R) = {g ∈ Γ|d(g(0), 0) ≤ R}.
With 0, v1, and v2 as three points in R2, we can draw a parallelogram P with these points as vertices.
Let d > 0 be the length of the longer diagonal of P .
(1) Show that for any g ∈ Γ(R), g(P ) is contained in the closed ball BR+d(0).
(2) Show that for R > d,

g∈Γ(R) g(P ) ⊇ BR−d(0).
(3) From (1) and (2), derive that pi(R − d)2 ≤ |Γ(R)| · Area(P ) ≤ pi(R + d)2 for all R > d, where |Γ(R)|
is the cardinality of Γ(R), then find lim
R→∞
|Γ(R)|/R2.
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