## Gromov norm and Turaev-Viro invariants of 3-manifolds. (arXiv:1705.09964v3 [math.GT] UPDATED)

We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants \$TV_r\$ and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold \$M\$, with (possibly empty) toroidal boundary, \$\log |TV_r (M)|\$ is bounded above by a function linear in \$r\$ and whose slope is a positive universal constant times the Gromov norm of \$M\$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of \$6j\$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has \$\log| TV_r (M)|\geqslant B \ r\$, for some \$B&gt;0\$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose \$SO(3)\$ Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [DK:AMU] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class gro查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants \$TV_r\$ and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold \$M\$, with (possibly empty) toroidal boundary, \$\log |TV_r (M)|\$ is bounded above by a function linear in \$r\$ and whose slope is a positive universal constant times the Gromov norm of \$M\$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of \$6j\$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has \$\log| TV_r (M)|\geqslant B \ r\$, for some \$B>0\$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose \$SO(3)\$ Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [DK:AMU] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class gro