Related papers: On the torsion in symmetric powers on congruence s…

In [MzM] we defined a regularized analytic torsion for quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behaviour of the analytic torsion as the lattices…

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an…

We study the homology group of the Milnor fiber boundary of a hyperplane arrangement in $\mathbb{C}^{3}$. By the work of N\'emethi--Szil\'ard, the homeomorphism type of the Milnor fiber boundary is combinatorially determined, and an…

According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been…

We compute the twisted cohomology of the mapping class group with level structures, with coefficients in the $r$-tensor powers of the Prym representations for any positive integer $r$. When $r\ge 2$, we show that the cohomology exhibits…

The signature of a closed manifold is an important geometric topology. Let $M$ be a closed manifold and $N$ be a codimension 2 submanifold of it. Given certain homotopy conditions, Higson, Xie and Schick proved an invariance theorem in…

Let $\Sigma_{g,n}$ be a compact oriented surface with genus $g\geq 2$ bordered by $n$ circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan symplectic form on the space of…

We consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values $\lambda_{tr}$. The invariant descends to an invariant of elements of $\mathcal{B}_n\diagup\mathcal{Z}_n$, the braid group modulo…

We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in $\mathbb{R}^d$. Our bound is explicit and captures the correct order of growth in $d$, improving upon the existing…

The twisted suspension of a manifold is obtained by surgery along the fibre of a principal circle bundle over the manifold. It generalizes the spinning operation for knots and preserves various topological properties. In this article, we…

We give a new proof of a conjecture of Schinzel on the intersection of a subvariety of codimension at least 2 in a power of the multiplicative group with a torus of dimension 1. The proof rests on a geometric B\'ezout's theorem of P.…

In this paper, as a continuation of [30], we consider the Gromov-Hausdorff convergence and collapsing in the family of compact Riemannian manifolds with boundary satisfying lower bounds on the sectional curvatures of interior manifolds,…

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $\Gamma \times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $\Gamma$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq…

We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known…

We consider the Reidemeister torsion associated with SL(2, C)-representations of a knot group. A bifurcation point in the SL(2, C)-character variety of a knot group is a character which is given by both an abelian SL(2, C)-representation…

Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here n is the…

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the…

Rank-one symmetric spaces carry a solvable group model which have a generalization to a larger class of Lie groups that are one-dimensional extensions of nilpotent groups. By examining some metric properties of these symmetric spaces, we…

We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a…

We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of…