Related papers: 6j-symbols, hyperbolic structures and the Volume C…

Let G be a graph of hyperbolic groups with 2-ended edge groups. We show that G is hierarchically hyperbolic if and only if G has no distorted infinite cyclic subgroup. More precisely, we show that G is hierarchically hyperbolic if and only…

Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant $\alpha >0$ such that for every finite subset $U$ that is not contained in a virtually cyclic…

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized…

We define and study two structures associated to permutation groups: Dirichlet characters on permutation groups, and the "cycle form," a bilinear form on the group algebras of permutation groups. We use Dirichlet characters and the cycle…

We define a family of Turaev-Viro type invariants of hyperbolic $3$-manifolds with totally geodesic boundary from the $6j$-symbols of the modular double of $\mathrm U_{q}\mathfrak{sl}(2;\mathbb R)$, and prove that these invariants decay…

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special…

We study topological properties of automorphisms of a 6-dimensional torus generated by integer matrices symplectic with respect to either the standard symplectic structure in six-dimensional linear space or a nonstandard symplectic…

We study the infimum of the renormalized volume for convex-cocompact hyperbolic manifolds, as well as describing how a sequence converging to such values behaves. In particular, we show that the renormalized volume is continuous under the…

The corrected triple sum expression of Ali\v{s}auskas (1987) for the recoupling (Racah) coefficients (6j-symbols) of the symmetric (most degenerate) representations of the orthogonal groups SO(n) (previously derived from the fourfold sum…

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the…

Let $M$ be a closed $n$-manifold with nonzero simplicial volume. A central result in systolic geometry from Gromov is that systolic volume of $M$ is related to its simplicial volume. In this short note, we show that systolic volume of…

Recently presented explicit formulae for asymptotic expansions of Feynman diagrams in the Sudakov limit are applied to typical two-loop diagrams. For a diagram with one non-zero mass these formulae provide an algorithm for analytical…

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural…

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every…

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves…

In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.

We study the asymptotic behaviour of the cohomology of subgroups $\Gamma$ of an algebraic group $G$ with coefficients in the various irreducible rational representations of $G$ and raise a conjecture about it. Namely, we expect that the…

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the…

We compute the real part of the semi-classical limit of the sequence of quantum hyperbolic invariants (QHI) of the figure-eight knot complement $M$. We show that it is rigid, in the sense that it does not depend on the choice of holonomy…