For each $g\ge 2$, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL$_2(\mathbb{C})$ whose image is contained in the fundamental group of some arithmetic hyperbolic $3$-manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.
We prove that closed arithmetic hyperbolic $n$-manifolds of bounded rank have bounded adjoint trace field degree, and consequently have lower bounded injectivity radius. From this, we obtain a number of corollaries. In particular, we obtain that closed arithmetic hyperbolic $3$-manifolds of bounded rank are contained i...
We construct finite covers of a fixed product of two closed hyperbolic surfaces of genus two whose fundamental groups are generated by at most fifteen elements and whose injectivity radii tend to infinity. The construction uses fibre products over finite groups. These covers are closed aspherical four-manifolds with un...
We show that for any $g\geq2$, there exist uncountably many non-isomorphic smooth affine threefolds that are Stein deformation equivalent to $T^\ast\Sigma_g\times\mathbb{C}$, the product of the cotangent bundle of a genus $g$ oriented surface with the complex plane, answering a question of Ivan Smith. In fact, we prove...
Let $f \colon S \rightarrow B$ be a fibred surface over an algebraically closed field $k$ of characteristic $p \geq 5$, where $S$ is a minimal smooth projective surface of general type and $B$ is a smooth projective curve of genus $b\geq 2$. We prove that the group of fibration-preserving automorphisms of $f$ has order...
Let $k$ be an algebraically closed field of characteristic $p>0$, and let $S$ be a minimal smooth projective surface of general type over $k$. If the second Chern class satisfies $c_2(S)<0$, then the order of the automorphism group satisfies $|\mathrm{Aut}_k(S)|<2598(K_S^2)^4$, and the order of every abelian subgroup $...
Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $\Sigma$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^*(X_{\Sigma})$ of the associated toric variety $X_{\Sigma}$ is isomorphic to a permutation representation of...
Tao Gui, Chu-Shi Qin, Kai-Zhe Shen et al.· 0 citations
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