Asymptotics of analytic torsion for hyperbolic three-manifolds
In this heavy technical article, the author proves limit theorems for a sequence of regularized analytic torsions on certain non-compact but finite-volume hyperbolic $3$-manifolds. More precisely, let $M_n$ be a sequence of certain finite-volume hyperbolic $3$-manifolds (i.e. arise by dividing the hyperbolic $3$-space with appropriate torsion-free lattices), being all diffeomorphic to $T^2 \times [1, +\infty )$ where $T^2$ is the flat $2$-torus, satisfying various kind of technical conditions listed up in Section 1.3 in the article. Let $V=V(n_1, n_2)$ with $n_1\not= n_2$ be the irreducible representation on $S^{n_1}{\mathbb C}^2\otimes S^{n_2}\overline{\mathbb{C}}^2$ of $\mathrm{SL}(2,\mathbb{C})$. Consider the $V$-valued \textit{regularized analytic torsion} $T_R(M_n;V)$ of $M_n$ regarded as a quotient of the hyperbolic $3$-space; likewise, let $T^{(2)}(M_n;V)$ denote the $V$-valued \textit{$L^2$-analytic torsion} of $M_n$ and put $t^{(2)}(V):=\frac{\log T^{(2)}(M_n;V)}{\mathrm{vol} M_n}$ for its universal (i.e. independent of $n$) term identified with the $L^2$-analytic torsion of the hyperbolic $3$-space itself (for their precise definitions based on the Selberg trace formula cf. Section 5.3 in the article). The author's main result is that \[ \lim_{n\rightarrow +\infty}\frac{\log T_R(M_n;V)}{\mathrm{vol} M_n} =t^{(2)}(V) \] (see Theorem A or Theorem 5.1 in the article) together with a variant (see Theorem B or Theorem 6.2) and a corollary (see Proposition C in the article) of this statement.
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