Rational growth in torus bundle groups of odd trace

Profinite genus of fundamental groups of torus bundles

In this paper we establish lower and upper bounds for the cardinality of the profinite genus of the fundamental group $π_{1}(M_A)\cong (\mathbb{Z} \times \mathbb{Z})\rtimes_{A}\mathbb{Z}$ of a torus bundle $M_{A}$ in terms of the number of ideal classes of the order $\mathbb{Z}[λ]$, where $λ$ is an eigenvalue of the matrix $A$ in $\mathrm{GL}_{2}(\mathbb{Z})$.

Descent of line bundles to GIT quotients of flag varieties by maximal torus

Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.

Classification of equivariant vector bundles over two-torus

We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six points are sufficient to classify equivariant vector bundles except a few cases. To do it, we calculate homotopy of the set of equivariant clutching maps. And, classification on real projective plane, Klein bottle will appear soon

穿孔环面丛上的非标准双向序

Non-standard bi-orders on punctured torus bundles

本文研究穿孔环面丛群的非标准双向序，证明其最大及次大真凸子群与标准双向序一致，并确认这些群确实存在非标准双向序。

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, $T_ϕ$, over the circle with pseudo-Anosov monodromy $ϕ$, there are associated two tessellations of the complex plane: one, $Δ(ϕ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(ϕ)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $Δ(ϕ)$ and $CW(ϕ)$.

Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

Given a matrix $A\in SL(N,\Z)$, form the semidirect product $G=\Z^N\rtimes_A \Z$ where the $\Z$ factor acts on $\Z^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.