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In this paper, we generalize the Cosmetic Surgery Conjecture to an $n$-cusped hyperbolic $3$-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For $n=1$ and $2$, we show them without the assumption.

Update: The Cosmetic Surgery Conjecture modulo finitely many Dehn-filling coefficients has been a well-known classical result, so the first main result of this paper is not new. (But the author was initially unaware of this fact, and the tools and techniques used here are very different from all the classically known methods.) The second main result of the paper, that is, the generalized Cosmetic Surgery Conjecture for the 2-cusped case is new, but superseded by the author's later work.

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field, and let $A$ be any quaternion algebra over $K$ such that $A\otimes _K\mathbb {R}\cong M_2(\mathbb {R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1

Assuming Lehmer's conjecture, we estimate the degree of the trace field $K(M_{p/q})$ of a hyperbolic Dehn-filling $M_{p/q}$ of a 1-cusped hyperbolic 3-manifold $M$ by $$ \dfrac{1}{C}(\max\;\{|p|,|q|\})\leq \text{deg }K(M_{p/q}) \leq C(\max\;\{|p|,|q|\}) $$ where $C=C_M$ is a constant that depends on $M$.

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

Abstract Assuming Lehmer's conjecture, we estimate the degree of the trace field of a hyperbolic Dehn filling of a 1‐cusped hyperbolic 3‐manifold by where is a constant that depends on .

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $ε&gt;0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x) &gt; (\text{vol}(M_i))^{\tfrac{1}{4}-ε}$. We get similar results in the

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a manifold has linearly independent cusp shapes, we show that the manifold has the desired property.To prove the results, we use the proof of the Bounded Height Conjecture in arithmetic

Let $M$ be an $n$-cusped hyperbolic $3$-manifold having rationally independent cusp shapes and $X$ be its holonomy variety. We first show that every maximal anomalous subvariety of $X$ containing the identity is its subvariety of codimension $1$ which arises by having a cusp of $M$ complete. Second, we prove if $X^{oa} =\emptyset$ , then $M$ has cusps which are, keeping some other cusps of it complete, strongly geometrically isolated from the rest. Third, we resolve the Zilber-Pink conjecture fo

Given a closed hyperbolic 3-manifold M, we construct a tower of covers with increasing Heegaard genus and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree.Using similar methods, we prove that for any > 0 there exist infinitely many congruence covers {M i } such that, for any x ∈ M, M i contains an embedded ball B x (with center x) satisfying vol B x > (vol M i ) 1/4-.We get similar results for an arithmetic noncompact case.

Let $M$ be a $2$-cusped hyperbolic $3$-manifold. By the work of Thurston, the product of the derivatives of the holonomies of core geodesics of each Dehn filling of $M$ is an invariant of it. In this paper, we classify Dehn fillings of $M$ with sufficiently large coefficients using this invariant. Further, for any given two Dehn fillings of $M$ (with sufficiently larger coefficients), if their aforementioned invariants are the same, it is shown their complex volumes are the same as well.

Let $M$ be an $n$-cusped hyperbolic $3$-manifold having rationally independent cusp shapes and $X$ be its A-polynomial. We first show that every maximal anomalous subvariety of $X$ containing the identity is its subvariety of codimension $1$ which arises by having a cusp of $M$ complete. Second, we prove if $X^{oa} =\emptyset$ , then $M$ has cusps which are, keeping some other cusps of it complete, strongly geometrically isolated from the rest. Third, we resolve the Zilber-Pink conjecture for A-

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.

Abstract Let be a 2‐cusped hyperbolic 3‐manifold. By the work of Thurston, the product of the derivatives of the holonomies of core geodesics of each Dehn filling of is an invariant of it. In this paper, we classify Dehn fillings of with sufficiently large coefficients using this invariant. Further, for any given two Dehn fillings of (with sufficiently larger coefficients), if their aforementioned invariants are the same, it is shown their complex volumes are the same as well.