Bruhat-Tits Trees Research Articles

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

AbstractSarnak's conjecture in quantum computing concerns when the groups and over cyclotomic rings with , , are generated by the Clifford‐cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group has only if is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary groups in the families and , , by letting them act on Bruhat–Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families and , the corank grows doubly exponentially in as ; it is 0 precisely when , and indeed, the cyclotomic unitary groups are generated by torsion elements (in fact by Clifford‐cyclotomic gates) for these . We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section, we execute a program of Sarnak to show that our results for the and families are sufficient to give a second proof of Sarnak's conjecture.

Joint partial equidistribution of Farey rays in negatively curved manifolds and trees

Abstract We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

Probing holography in p -adic CFT

We holographically calculate the partition functions of certain types of isotropic sectors of the CFTs dual to Bruhat-Tits trees and $p$-adic BTZ black holes. Along the way, we propose new spectral decompositions of the Laplacian operator other than the plane-wave basis on these two types of background, with both analytical and numerical evidence. We extract the density of states and hence entropy from the BTZ partition function via the inverse Laplace transform. Then the one-loop Witten diagram is computed in the $p$-adic BTZ black hole background, yielding constraints on the heavy-heavy-light averaged three-point coefficient of its boundary $p$-adic CFT. Finally, for general $p$-adic CFTs (not necessarily holographic), we analyze the representation theory of their global conformal group $PGL\left(2,\mathbb{Q}_p\right)$, and discuss the suitability of different representations as Hilbert spaces of $p$-adic CFT.

Holographic tensor networks from hyperbolic buildings

We introduce a unifying framework for the construction of holographic tensor networks, based on the theory of hyperbolic buildings. The underlying dualities relate a bulk space to a boundary which can be homeomorphic to a sphere, but also to more general spaces like a Menger sponge type fractal. In this general setting, we give a precise construction of a large family of bulk regions that satisfy complementary recovery. For these regions, our networks obey a Ryu-Takayanagi formula. The areas of Ryu-Takayanagi surfaces are controlled by the Hausdorff dimension of the boundary, and consistently generalize the behavior of holographic entanglement entropy in integer dimensions to the non-integer case. Our construction recovers HaPPY-like codes in all dimensions, and generalizes the geometry of Bruhat-Tits trees. It also provides examples of infinite-dimensional nets of holographic conditional expectations, and opens a path towards the study of conformal field theory and holography on fractal spaces.

Ramanujan Complexes and Golden Gates in PU(3)

In a seminal series of papers from the 80’s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for \(GL_{2}\) (Deligne’s theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with \(SL_{2}({\mathbb {Q}}_{p})\). As a result, they obtained explicit Ramanujan Cayley graphs from \(PSL_{2}\left( {\mathbb {F}}_{p}\right) \), as well as optimal topological generators (“Golden Gates”) for the compact Lie group PU(2). In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for \(PU_{3}\) by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with \(SL_{3}({\mathbb {Q}}_{p})\) and \(SU_{3}({\mathbb {Q}}_{p})\), while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from \(PSL_{3}({\mathbb {F}}_{p})\) and \(PSU_{3}({\mathbb {F}}_{p})\), as well as golden gates for PU(3).

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Consider an infinite homogeneous tree \mathcal{T}_n of valence n+1 , its group Aut (\mathcal{T}_n) of automorphisms, and the group Hier (\mathcal{T}_n) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of \mathcal{T}_n that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut (\mathcal{T}_n) is spherical in Hier (\mathcal{T}_n) , i.e., any irreducible unitary representation of Hier (\mathcal{T}_n) contains at most one Aut (\mathcal{T}_n) -fixed vector. We present a combinatorial description of the space of double cosets of Hier (\mathcal{T}_n) with respect to Aut (\mathcal{T}_n) and construct a “new” family of spherical representations of Hier (\mathcal{T}_n) . We also show that the Thompson group Th has PSL (2,\mathbb{Z}) -spherical unitary representations.

Nonarchimedean holographic entropy from networks of perfect tensors

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.

On the nonarchimedean quadratic Lagrange spectra

We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees.

Zeta and L-functions of finite quotients of apartments and buildings

In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G =PGL3 and PGSp4 over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara’s theorem for finite quotients of the Bruhat–Tits trees. This identity is shown to agree with the q = 1 version of the analogous identities for finite quotients of the building of G established in [KL14, KLW10, FLW13], verifying the philosophy of the field with one element by Tits. A new identity for finite quotients of the building of PGSp4 involving the standard L-function (Theorem 6.3), complementing the one in [FLW13] which involves the spin L-function, is also obtained.

Tree-Lattice Zeta Functions and Class Numbers

We extend the theory of Ihara zeta functions to noncompact arithmetic quotients of Bruhat–Tits trees. This new zeta function turns out to be a rational function despite the infinite-dimensional setting. In general, it has zeros and poles in contrast to the compact case. The determinant formulas of Bass and Ihara hold if we define the determinant as the limit of all finite principal minors. From this analysis we derive a prime geodesic theorem, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.

On optimal embeddings and trees

We apply the theory of Bruhat–Tits trees to the study of optimal embeddings from orders of rank two and three to quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in an indefinite quaternion algebra yield optimal representations of such orders. This completes previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as suborders. The same technique is used in the second part of this work to compute local embedding numbers, which extends previous results by J. Brzezinski.

Green’s functions on Mumford curves

We study an analogue of a Green’s function on Mumford curves. The meromorphic continuation of this Green’s function comes from a “differential equation,” and we interpret the special value at \(s=0\) as a “volume” of the corresponding curve. Using harmonic analysis on Bruhat-Tits trees, we derive a “Kronecker limit formula” for the Green’s functions in question, which connects the derivative at \(s=0\) with the Manin–Drinfeld theta functions. Following Gross’ description of Neron’s local height pairing on Mumford curves, the special derivative here then equals twice of the Neron’s local height with sign changed.

Effective mixing and counting in Bruhat–Tits trees

Let ${\mathcal{T}}$ be a locally finite tree, $\unicode[STIX]{x1D6E4}$ be a discrete subgroup of $\,\operatorname{Aut}\,({\mathcal{T}})$ and $\widetilde{F}$ be a $\unicode[STIX]{x1D6E4}$-invariant potential. Suppose that the length spectrum of $\unicode[STIX]{x1D6E4}$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map $\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}\rightarrow \unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}$ with respect to the measure $m_{\unicode[STIX]{x1D6E4},F}^{\unicode[STIX]{x1D708}^{-},\unicode[STIX]{x1D708}^{+}}$ under the assumption that $\unicode[STIX]{x1D6E4}$ is full and $(\unicode[STIX]{x1D6E4},\widetilde{F})$ has a weighted spectral gap. We also obtain the effective formula for the number of $\unicode[STIX]{x1D6E4}$-orbits with weights in a Bruhat–Tits tree ${\mathcal{T}}$ of an algebraic group.

Graphs of Hecke operators

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.

On the characteristic p valued measure associated to Drinfeld discriminant

In this paper, after reviewing known results on functions over Bruhat-Tits trees and the theory of characteristic $p$ valued modular forms, we present some structure of the tempered distributions on the projective space ${\P1}(\bk_\infty)$ over a complete function field $\bk_\infty$ of characteristic $p$, and calculate the characteristic $p$ valued measure associated to the Drinfeld discriminant and the characteristic $p$ valued measure associated to the Poinca\'e series.

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of ℚ p and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of p-adic geometry, to encode a dendrogram X in a p-adic field K means to fix a set S of K-rational punctures on the p-adic projective line ℙ1. To ℙ1 \ S is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers X, a method first used by F. Kato in 1999 in the classification of discrete subgroups of PGL2(K). Next, we show how the p-adic moduli space \( \mathfrak{M}_{0,n} \) of ℙ1 with n punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on ℙ1. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a p-adic algebraic curve with totally degenerate reduction modulo p. Finally, we indicate some of our results in the study of general discrete actions on ℙ1, and their relation to p-adic Hurwitz spaces.

A logarithm law for automorphism groups of trees

Let Γ be a geometrically finite tree lattice. We prove a Khintchine-Sullivan type theorem for the Hausdorff measure of the set of points at infinity of the tree that are well approximated by the parabolic fixed points of Γ. Using Bruhat-Tits trees, an application is given for the Diophantine approximation of formal Laurent series in the variable X −1 over the finite field $${\mathbb{F}_{q}}$$ by rational fractions in X over $${\mathbb{F}_{q}}$$ satisfying some congruence properties.

ON COMBINATORIAL ANALOGS OF THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE

The goal of this article is to construct and study groups which, from the point of view of the theory of representations, should resemble the group of diffeomorphisms of the circle. The first type of such groups are the diffeomorphism groups of -adic projective lines. The second type are groups consisting of diffeomorphisms (satisfying certain conditions) of the absolutes of Bruhat-Tits trees; they can be regarded as precisely the diffeomorphism groups of Cantor perfect sets. Several series of unitary representations of these groups are constructed, including the analogs of highest-weight representations.

Non-archimedean strings and Bruhat-Tits trees

It is shown that the Bruhat-Tits tree for thep-adic linear groupGL(2) is a natural non-archimedean analog of the open string world sheet. The boundary of the tree can be identified with the field ofp-adic numbers. We construct a “lattice” quantum field theory on the Bruhat-Tits tree with a simple local lagrangian and show that it leads to the Freund-Olson amplitudes for emission processes of the particle states from the boundary.

A NOTE ON MULTILOOP CALCULUS IN P-ADIC STRING THEORY

The technique for finding correlation functions on homogeneous spaces of PGL(ℚp) groups (factorized Bruhat-Tits trees Tp/ΓN with finite number of cycles) is presented. It was shown in Refs. 5 and 6 that the homogeneous spaces Tp/ΓN are in fact the multiloop world sheets in p-adic string theory.