Related papers: 6j-symbols, hyperbolic structures and the Volume C…
We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows…
We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics…
The details are expounded of an old treatment of the limit of the su(N) structure constants as N tends to infinity. A recently derived parity property of the series expansion is shown to be the same as the known mirror symmetry of 6-j…
Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the…
We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…
A classical result of J\o rgensen and Thurston shows that the set of volumes of finite volume complete hyperbolic $3$-manifolds is a well-ordered subset of the real numbers of order type~$\omega^\omega$; moreover, each volume can only be…
We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We also consider the more general situation…
We adapt the Gurau's proof (2008) about the asymptotic limit of Ponzano-Regge formula to supersymmetric 6jS symbols according to their intrinsic parities alpha, beta, gamma. The behaviour at a large scaling shows significant differences…
We study a class of two dimensional partially hyperbolic systems, not necessarily skew products, trying to establish the germ of a general theory. To illustrate the scope of the theory, we apply our results to the case of fast-slow…
In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps…
The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in $S^3$. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of…
We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.
For a hyperbolic link K in the thickened torus with no bigons, we show that there is a decomposition of the complement of a link L, obtained from augmenting K, into torihedra. We further decompose the torihedra into angled pyramids and…
A family of non-radial solutions of the Yamabe equation, with reference the hyperbolic space, is constructed using power series. As a result, we obtain a family of asymptotically hyperbolic metrics, with spherical conformal infinity, with…
Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element.…
We consider the universal solution of the Gervais-Neveu-Felder equation in the ${\cal U}_q(sl_2)$ case. We show that it has a quasi-Hopf algebra interpretation. We also recall its relation to quantum 3-j and 6-j symbols. Finally, we use…
When does the amount of torsion in the homology of an arithmetic group grow exponentially with the covolume? We give many examples where this is so, and conjecture precise conditions.
The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than…
The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod…
We determine the growth of the dimension of the slope subspaces of the cohomology of arithmetic subgroups in reductive algebraic groups as a function of the slope.