Teichmüller Theory and Applications to Geometry, Topology, and Dynamics
Volume 1: Teichmüller Theory(Second Printing)

Errata, complete as of June 16, 2018

We thank Wolf Jung, Didac Martinez-Granado, Ahmad Rafiqi, SamuelRoth, Xiaoguang Wang, and Alex Wright for their contributions to thislist. Our apologies to anyone whose name has been inadvertently omitted.
New errata are marked in red.
ErrataPage 21 Exercise 1.8.11: Part 1 should read “Two nonidentity Möbiustransformations commute if and only if they have the same fixed points,
or are commuting involutions each interchanging the fixed points of theother.” (As the statement currently stands, f (z) = z and g(z) = 1/z area counterexample.)
Page 155[posted June 16, 2018] Equation 4.7.9 should not have an
inverse on the left side; it should be(id µL)h = µ.

4.7.9

Page 222 [posted September 29, 2017] Equation 5.4.9 should have a pthroot on the right:✓Z◆1/p|q|p
kqkp.
2p 2
X ⇢
Page 267[posted March 25, 2017] Equation 6.6.7: The second term
should be “(1 + t@⇠ + o(t))”, not “(1 + @⇠ + o(t))”.
Page 268

Line 3: “@⇠ = 0” should be “@⇠ = ⌫”.

Page 272 [posted June 16, 2018] Equation 6.6.30: On the left side, theµ in the numerator should be sµ:✓◆dsµdwµdw=.
6.6.30ds 1 t(t + s)|µ|2 dw
1 t2 |µ|2 dw
s=0Page 294 [posted March 25, 2017] Proposition 6.12.4: in the second lineof part 2, “a an open subset” should be “an open subset”.
Page 385[posted Jan. 22, 2016] In Definition A7.2.1, after “given bythe formula” the subscript and superscript on ⇢ (after the sum) should beexchanged; it should be
k+1X
i=0notU \···\Uk+1( 1)i ⇢U0 \···Ub ···\U
0k+1X
i=0

ik+1b ···\UU \···U

ik+1( 1)i ⇢U00 \···\U
k+1

Our convention (equation A7.1.1) is that when V ⇢ U , we write ⇢VU , not thereverse: the small set goes on top. On top, we are taking the intersectionof more sets, so it is smaller.
Page 392 Third paragraph of Section A7.4, next to last line:⇤⇤
H 1 (X, OX), not H(X, OX)
Page 392

Equation A7.4.1:'U11 'U2 : C ⇥ (U1 \ U2 ) ! C ⇥ U1

should be'U11 'U2 : (U1 \ U2 ) ⇥ C ! U1 ⇥ C
Equation A7.4.2:'U11 'U2 (z, x) = MU1 ,U2 (x)z, x
should be'U11 'U2 (x, z) = x, MU1 ,U2 (x)zPage 412[new Dec. 22, 2017] In Theorem A9.14 (Serre duality) weshould have said that X is of dimension n. On the fourth line of thetheorem, at the very end, ⌦ should be ⌦X . Thus the theorem should read:
Let V be an analytic vector bundle on a compact complex manifold Xof dimension n, and let V ⇤ be the dual vector bundle. Then the pairn,n king of A0,k(V ⇤ ) induces a duality of H k X, OX (V ) withX (V ) with DX⌦nn k⇤
HX, OX (V ) ⌦ ⌦X .
Notes and amplifications
Pages 267 and 268 [added March 25, 2017] Proof of Lemma 6.6.3: Areader asked, “why is ker[D S (⌧ )] spanned by tangent vectors of curves in11S ( S (⌧ ))? Answer: by Theorem 6.5.1, part 2, S ( S (⌧ )) is a manifold,
and the tangent space is always spanned by tangent vectors to curves in themanifold. This depends on S being a split submersion; see Proposition
A5.6 and the discussion page 363 following Definition A5.7.
The reader also asked, how is the assumption that ⇠ vanishes on R being
used? Answer: It is used to say (in equation 6.6.8) that when z 2/ H, i.e.,
when z 2 H⇤ , thendf t@⇠ (z) = z + o(t).

(This also uses Proposition 4.7.6.)