Invariant Trace Field Research Articles

Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We show that there are constants $A,B$ such that every such manifold $M$ is homotopy equivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices, all of degrees bounded by $B$.

Geometry of biperiodic alternating links

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.

Decreasing height along continued fractions

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map$x\mapsto x^{-1}-\lfloor x^{-1}\rfloor$eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of non-cocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that non-cocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of non-negative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

Realizing algebraic invariants of hyperbolic surfaces

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.

Systoles of arithmetic hyperbolic surfaces and $3$-manifolds

Our main result is that for any positive real number x0, the set of commensurability classes of arithmetic hyperbolic 2– or 3–manifolds with fixed invariant trace field k and systole bounded below by x0 has density one within the set of all commensurability classes of arithmetic hyperbolic 2– or 3–manifolds with invariant trace field k. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is Q, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound to instead grow with the area/volume. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser–Sarnak and Katz–Schaps–Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3–manifolds with a small area totally geodesic surface.

On the arithmetic dimension of triangle groups

Let $\Delta=\Delta(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $\pi/a,\pi/b,\pi/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $\Delta$ to be the number of split real places of the quaternion algebra generated by $\Delta$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.

Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister–Schreier algorithm as implemented in magma to generate these surfaces.

Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds

Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3-manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number $k$ manifold it is enough to choose $3k$ specific parameters. For many hyperbolic link complements, this approach allows one to compute the field from a link diagram. We also give examples of infinite families of links where a single parameter can be chosen to generate the field, and the polynomial for it can be constructed from the link diagram as well.

A note on trace fields of complex hyperbolic groups

We show that if \Gamma is an irreducible subgroup of SU(2,1), then \Gamma contains a loxodromic element A . If A has eigenvalues \lambda_1 = \lambda e^{i\varphi} , \lambda_2 = e^{-2i\varphi} , \lambda_3 = \lambda^{-1}e^{i\varphi} , we prove that \Gamma is conjugate in SU(2,1) to a subgroup of SU (2,1,\mathbb{Q}(\Gamma,\lambda)) , where \mathbb{Q}(\Gamma, \lambda) is the field generated by the trace field \mathbb{Q}(\Gamma) of \Gamma and \lambda . It follows from this that if \Gamma is an irreducible subgroup of SU(2,1) such that the trace field \mathbb{Q}(\Gamma) is real, then \Gamma is conjugate in SU(2,1) to a subgroup of SO(2,1). As a geometric application of the above, we get that if G is an irreducible discrete subgroup of PU(2,1), then G is an \mathbb R -Fuchsian subgroup of PU(2,1) if and only if the invariant trace field k(G) of G is real.

The extended Bloch group and algebraic K-theory

Abstract We define an extended Bloch group for an arbitrary field F, and show that this group is naturally isomorphic to K 3 ind ( F ) $K_3^{\operatorname{ind}}(F)$ if F is a number field. This gives an explicit description of K 3 ind ( F ) $K_3^{\operatorname{ind}}(F)$ in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class [ M ] $[M]$ in K 3 ind ( k ) ⊗ ℤ [ 1 2 ] $K_3^{\operatorname{ind}}(k)\otimes \mathbb {Z}[\frac{1}{2}]$ . If M is closed, 2 [ M ] $2[M]$ is well defined in K 3 ind ( k ) $K_3^{\operatorname{ind}}(k)$ , and if M has cusps, 8 [ M ] $8[M]$ is well defined in K 3 ind ( k ) $K_3^{\operatorname{ind}}(k)$ .

The quantum content of the gluing equations

The gluing equations of a cusped hyperbolic 3‐manifold M are a system of polynomial equations in the shapes of an ideal triangulation T of M that describe the complete hyperbolic structure of M and its deformations. Given a Neumann‐Zagier datum (comprising the shapes together with the gluing equations in a particular canonical form) we define a formal power series with coefficients in the invariant trace field of M that should (a) agree with the asymptotic expansion of the Kashaev invariant to all orders, and (b) contain the nonabelian Reidemeister‐Ray‐Singer torsion of M as its first subleading “1‐loop” term. As a case study, we prove topological invariance of the 1‐loop part of the constructed series and extend it into a formal power series of rational functions on the PSL.2;C/ character variety of M . We provide a computer implementation of the first three terms of the series using the standard SnapPy toolbox and check numerically the agreement of our torsion with the Reidemeister‐Ray‐Singer for all 59924 hyperbolic knots with at most 14 crossings. Finally, we explain how the definition of our series follows from the quantization of 3‐dimensional hyperbolic geometry, using principles of topological quantum field theory. Our results have a straightforward extension to any 3‐manifold M with torus boundary components (not necessarily hyperbolic) that admits a regular ideal triangulation with respect to some PSL.2;C/ representation.

Bloch invariants of hyperbolic 3-manifolds

We define an invariant \beta(M) of a finite volume hyperbolic 3-manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. \beta(M) lies in a subgroup of \B(\C) of finite \Q-rank determined by the invariant trace field of M. Moreover, the Chern-Simons invariant of M is determined modulo rationals by \beta(M). This leads to a simplicial formula and rationality results for the Chern Simons invariant which appear elsewhere. Generalizations of \beta(M) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of B(C).

Commensurability and the character variety

Recall that hyperbolic 3-manifolds M and N are said to be commensurable if they have a common finite sheeted covering. This is equivalent to the fundamental groups having subgroups of finite index which are conjugate in PSL(2,C). In general it is very difficult to determine if two manifolds are commensurable or not, once the most obvious invariants of commensurability (for example, the invariant trace field, see [13] and [17]) agree. When M is a finite volume hyperbolic 3-manifold with a single cusp, its SL(2,C)-representation and character varieties, denoted respectively, by R(M) and X(M) throughout, have been fundamental tools in understanding the topology of M , see [6], [5], and [4]. These techniques can be extended to the PSL(2,C)-character variety of M , which we denote by Y (M) ([2], and see §2.1 for some details). Throughout, for either SL(2) or PSL(2), we use the subscript 0 to denote a component of X(M) (or Y (M)) containing the character of a faithful discrete representation of π1(M). The main results of this paper concern how Y0(M) be can used to detect incommensurability. For example, one of the main results can be summarized in the following (for terminology and definitions see §2):

Invariant Trace-Fields and Quaternion Algebras of Polyhedral Groups

Let P be a polyhedron in H3 of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of Isom H3. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group. As discussed in [12] and [13] for example (see §2 below), associated to a Kleinian group Γ of finite covolume is a pair (AΓ, kΓ) which is an invariant of the commensurability class of Γ; kΓ is a number field called the invariant trace-field, and AΓ is a quaternion algebra over kΓ. It has been of some interest recently (cf. [13, 16]) to identify the invariant trace-field and quaternion algebra associated to a Kleinian group Γ of finite covolume since these are closely related to the geometry and topology of H3/Γ. In this paper we give a method for identifying these in the case of polyhedral groups avoiding trace calculations. This extends the work in [15] and [11] on arithmetic polyhedral groups. In §6 we compute the invariant trace-field and quaternion algebra of a family of polyhedral groups arising from certain triangular prisms, and in §7 we give an application of this calculation to construct closed hyperbolic 3-manifolds with ‘non-integral trace’.

Commensurability and virtual fibration for graph manifolds

Two manifolds are commensurable if they have diffeomorphic finite covers. We would like invariants that distinguish manifolds up to commensurability. A collection of such commensurability invariants is complete if it always distinguishes non-commensurable manifolds. Commensurability invariants of hyperbolic 3-manifolds are discussed in [l]. The two main ones are the invariant trace field and the invariant quaternion algebra. The latter is a complete commensurability invariant in the arithmetic case, but not in general. The set of primes at which traces fail to be integral is another commensurability invariant, and examples are given in Cl] where the invariant quaternion algebras agree but this set does not. Another commensurability invariant discussed in [l] is the collection of “cusp fields” (the fields generated by cusp parameters). Craig Hodgson has pointed out that the set of PSL(2, Q)-classes of cusp parameters is a finer commensurability invariant than the cusp fields when the degree of some cusp field exceeds 3. Here we discuss commensurability of nonhyperbolic 3-manifolds. For 3-manifolds with geometric structure the classification is known (cf. Section 2): THEOREM A. For each of the six “Seifert geometries” S3, E3, S2 x IE’, E-U2 x [El, Nil, and PSL there is just one commensurability class of compact geometric 3-manifolds with the given structure (two for the last two geometries if orientation-preserving commensurability of oriented manifolds is considered). For the remaining nonhyperbolic geometry Sol, the commensurability classes are in one-one correspondence with real quadratic number jelds (such a manifold is covered by a torus bundle over a circle and the field in question is the field generated by an eigenvalue of the monodromy of this bundle). Noncompact finite volume nonhyperbolic geometric 3-manifolds admit geometric structures of both types W2 x lE3 and PSL and form just one commensurability class, also in the oriented case. In Section 3 we define several multiplicative invariants for prime 3-manifolds. A multiplicative invariant is one that multiplies by degree for covering spaces. Our invariants are most interesting for graph manifolds. Since the ratio of two multiplicative invariants is a commensurability invariant, we get several commensurability invariants also. The next two theorems, which use two of these invariants, are a start on the commensurability classification for manifolds with nontrivial geometric decomposition. Let M be a closed non-Seifert-fibered oriented 3-manifold obtained by pasting two Seifert manifolds, MI and M2, each having a torus as its boundary, along these tori. Suppose also that neither half Mi is the total space SMb of the circle bundle over the MSbius band with orientable total space (otherwise there is a double cover of M that either satisfies our requirements or is a torus bundle over the circle and is thus covered by Theorem A). To each

Invariants from triangulations of hyperbolic 3-manifolds

For any finite volume hyperbolic 3-manifold M we use ideal triangulation to define an invariant β(M) in the Bloch group B(C ). It actually lies in the subgroup of B(C ) determined by the invariant trace field of M . The Chern-Simons invariant of M is determined modulo rationals by β(M). This implies rationality and — assuming the Ramakrishnan conjecture — irrationality results for Chern Simons invariants.

Amalgamation and the invariant trace field of a Kleinian group

Let Γ be a Kleinian group of finite covolume and denote by Γ(2) the subgroup generated by {γ2:γ ∈ Γ}. In [9] the trace field of Γ(2) was shown to be an invariant of the commensurability class of Γ. In [8] this field was termed the invariant trace field of Γ and further properties of this field were studied. Following the notation of [8] we denote the invariant trace field of Γ by k(Γ).