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Commuting affine peripheral holonomies factor their logarithms on singular parameter spaces

Statement

Let $Z$ be a reduced complex analytic space with point $o$ , and suppose $u_{i}, v_{i}, a_{i}, b_{i}$ are holomorphic germs on $Z$ for $1 \leq i \leq k$ , with $k < \infty$ , $u_{i} (o) = v_{i} (o) = 0$ , $a_{i} (o) \neq 0$ and $τ_{i}^{0} = b_{i} (o) / a_{i} (o) \notin R$ . Suppose the affine maps $L_{i} (w) = e^{u_{i}} w + a_{i}, P_{i} (w) = e^{v_{i}} w + b_{i}$ commute at every parameter point. Then, after shrinking around $o$ , there are holomorphic germs $τ_{i}$ such that $v_{i} = τ_{i} u_{i}$ , $τ_{i} (o) = τ_{i}^{0}$ and $Im τ_{i} \neq 0$ throughout. In particular $u_{i} = 0$ if and only if $v_{i} = 0$ , and $v_{i} / u_{i} \to τ_{i}^{0}$ as the parameter tends to $o$ through $u_{i} \neq 0$ . No smoothness, dimension or rank hypothesis on $Z$ is required.

For $u_{i} \neq 0$ , there is a unique real pair $(p_{i}, q_{i})$ solving $p_{i} u_{i} + q_{i} v_{i} = 2 π i$ . With ${\tilde{u}}_{i} = \frac{2 π i}{p_{i} + τ_{i}^{0} q_{i}},$ one has $∣ {\tilde{u}}_{i} - u_{i} ∣ \leq \frac{∣ τ_{i} - τ_{i}^{0} ∣}{∣ Im τ_{i}^{0} ∣} ∣ u_{i} ∣ .$ Set ${\tilde{u}}_{i} = 0$ when $u_{i} = 0$ . The functions ${\tilde{u}}_{i}$ are continuous at such points, and $∥ \tilde{u} - u ∥ = o (∥ u ∥)$ as the parameter tends to $o$ through $u \neq 0$ .

Statement references

Petronio and Porti, Lemma 2.6 and Proposition 3.6

Facts & Assumptions

Given: The finite collection of holomorphic germs, their values at $o$ and the commuting affine maps in the Statement.

Proof

1.1givenconstructalgebra

If $k = 0$ , the collection of germs and every indexed assertion are empty. Both vectors are the unique vector of $C^{0}$ , with norm zero; there is no parameter with $u \neq 0$ , so the final limiting assertion is vacuous. This proves the Statement in that case. Henceforth suppose $k \geq 1$ . Composition of the two affine maps shows that commutativity is exactly $a_{i} (e^{v_{i}} - 1) = b_{i} (e^{u_{i}} - 1)$ . Let $E (t) = (e^{t} - 1) / t$ for $t \neq 0$ and $E (0) = 1$ ; its convergent series is $\sum_{n \geq 0} t^{n} / (n + 1)!$ , so it is holomorphic and nonzero near zero. Since $a_{i} (o) \neq 0$ , the holomorphic function $τ_{i} = (b_{i} / a_{i}) E (u_{i}) / E (v_{i})$ is defined near $o$ . The commutativity equation becomes exactly $v_{i} = τ_{i} u_{i}$ , and $τ_{i} (o) = b_{i} (o) / a_{i} (o)$ . The finitely many nonreal values permit one common neighborhood on which all $Im τ_{i}$ are nonzero. Thus all $τ_{i}$ are nonzero, proving the equivalence of their zero logarithms and the asserted quotient limit. The argument is an identity of holomorphic functions on $Z$ , so singular points cause no difficulty.

2.1step 1.1algebra

When $u_{i} \neq 0$ , real linear independence of $u_{i}$ and $v_{i} = τ_{i} u_{i}$ follows from $Im τ_{i} \neq 0$ . Hence the real linear map $(p, q) \mapsto p u_{i} + q v_{i}$ is invertible and gives the unique real pair. Its equation is $(p_{i} + q_{i} τ_{i}) u_{i} = 2 π i$ . Since $Im τ_{i}^{0} \neq 0$ , the denominator $p_{i} + q_{i} τ_{i}^{0}$ cannot vanish: if it did, its imaginary part would give $q_{i} = 0$ , then its real part $p_{i} = 0$ , contrary to the full-turn equation. Therefore ${\tilde{u}}_{i} / u_{i} = (p_{i} + q_{i} τ_{i}) / (p_{i} + q_{i} τ_{i}^{0})$ and $({\tilde{u}}_{i} - u_{i}) / u_{i} = q_{i} (τ_{i} - τ_{i}^{0}) / (p_{i} + q_{i} τ_{i}^{0})$ . The elementary bound $∣ p_{i} + q_{i} τ_{i}^{0} ∣ \geq ∣ q_{i} ∣ ∣ Im τ_{i}^{0} ∣$ proves the estimate, including $q_{i} = 0$ where its left side is zero.

3.1step 1.1step 2.1algebra∎

On a sufficiently small fixed neighborhood all $τ_{i}$ are bounded, so step 2.1 gives $∣ {\tilde{u}}_{i} ∣ \leq C ∣ u_{i} ∣$ . Thus extending ${\tilde{u}}_{i}$ by zero across $u_{i} = 0$ is continuous there; elsewhere its formula is continuous because it comes from an invertible real linear system. Put $η (z) = \max_{i} ∣ τ_{i} (z) - τ_{i}^{0} ∣ / ∣ Im τ_{i}^{0} ∣$ . Finiteness of $k$ and continuity give $η (z) \to 0$ as $z \to o$ . Applying the scalar estimates coordinatewise, also where $u_{i} = 0$ , gives $∥ \tilde{u} - u ∥ \leq η (z) ∥ u ∥$ . This proves the final assertion. No claim that $u$ is a coordinate system or has nonvanishing derivative has entered the argument.

Scope

The affine normalization retains a chosen peripheral fixed point or eigenline. This lemma supplies the logarithmic quotient and the filling-parameter comparison even when the shape space is singular. It supplies neither a holomorphic coordinate on an ordinary character quotient nor the analytic-space openness, isolated-complete-point, geometric realization or triangulation-existence prerequisites needed for general hyperbolic Dehn filling.

Used by

Augmented holonomy coordinates at a complete finite-volume hyperbolic structure Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Carlo Petronio and Joan Porti, Negatively Oriented Ideal Triangulations and a Proof of Thurston's Hyperbolic Dehn Filling Theorem (standard reference, not scraped)