Building lattices and zeta functions Anton Deitmar, Ming-Hsuan Kang & Rupert McCallum∗ Abstract: We give a Lefschetz formula for tree lattices and apply it to geometric zeta functions. We further generalize Bass’s approach to Ihara zeta functions to the higher dimensional case of a building. MSC: 51E24, 11M36, 20E42, 20F65, 22D05, 22E40 Contents 1 Affine buildings 3 1.1 Theautomorphismgroup . 3 1.2 Theboundary ........................... 8 1.3 Cuspidalflowandhorospheres . 11 1.4 Hyperbolicelements . 13 1.5 Levicomponents ......................... 17 2 The Lefschetz formula 18 arXiv:1412.3327v4 [math.GR] 5 Sep 2019 2.1 Periodicandweaklysymmetricbuildings . 18 2.2 Orbitalintegrals.......................... 20 2.3 StatementandproofoftheLefschetzformula . 21 2.4 Aseveralvariablezetafunction . 25 3 AGeometricapproachtozetafunctions 29 3.1 Bass’stranslationoperators . 29 3.2 Cocompactlattices . .. .. .. .. .. .. .. 32 ∗Funded by DFG grant DE 436/10-1 1 BUILDING LATTICES 2 3.3 Poincar´eseries........................... 34 3.4 Non-cocompactlattices. 39 Introduction In the nineteensixties, Yasutaka Ihara defined an analog of the Selberg zeta function for p-adic groups of split rank one. Later Jean-Pierre Serre observed that this zeta function can be defined for arbitrary finite graphs. It was an open question, whether the theory of the Ihara zeta function could be generalized to higher rank groups. This question has been answered affirmatively for p-adic groups in 2014 in the paper [17]. The corresponding generalization to arbitrary buildings is given in the current paper. Ihara provided in [22] the only known link between geometric and arithmetic zeta functions by showing that the Ihara zeta function for a finite arithmetic quotient of a Bruhat-Tits tree equals the Hasse-Weil zeta function of the corresponding Shimura curve. Generally, the Bruhat-Tits building of a p-adic group is the analogue of the symmetric space of a semi- simple Lie group. In the latter case a Lefschetz formula has been developed [1,8,10,15,24,25], which expresses geometrical data of the geodesic flow and its monodromy in terms of Lie algebra cohomology, or more general, foliation cohomology. This has been transferred to the case of p-adic groups in [9] and applied in [17]. The presentation in both papers is focused on the cohomological approach using the theory of reductive linear algebraic groups. In the present paper, we give a much simpler approach which entirely works in geometric terms and doesn’t use algebraic groups at all. Signifi- cantly, it is formulated with the automorphism group of a building instead. Thismakesthe papereasierto read, but leavesusin the curious situation of a Lefschetz formula without cohomology. We decided to call it a Lefschetz formula nevertheless because of its genealogy. This formula is also more general than its predecessor since it allows lattices which are not of Lie-type. A lattice Γ acting on a building X is said to be of Lie-type if the building is the Bruhat-Tits building of a p-adic linear group G and Γ′ ⊂ G for a finite index subgroup Γ′ of Γ. In general, the existence of non-Lie-type-lattices in this case is an open question. In the first part of the paper we prepare the necessary theory ofaffine buildings and their automorphism groups, starting from geometry and BUILDING LATTICES 3 moving towards group theory, as may be seen from the titles of the sub- sections. In the second part we develop the Lefschetz formula and give an application to a several variable zeta function which we define by an infinite sum over geometric terms and use the Lefschetz formula to show that it actually is a rational function. We get precise information on its sin- gularities in terms of spectral data. This is the higher rank generalization of the celebrated Ihara zeta function [22, 23, 31]. The third part is concerned with a different approach to the zeta func- tion which does not depend on the Lefschetz formula, but rather works like Bass’s approach to the Ihara zeta function in the rank one case [4]. We clarify its relation with the Lefschetz formula zeta function of the previous section and the Poincar´eseries. Finally, we formulate a conjecture concern- ing the rationality of the zeta function in case of a non-cocompact lattice. This problem has, in the rank one case, been solved in [18]. The results of this paper are applied in [19] to obtain prime geodesic theorems and, in their wake, results on class number asymptotics. For similar results in real or p-adic settings, see [11–13,30]. In order to illustrate what kind of class number asymptotics we mean, we give an example from the paper [18]: Theorem.(Class number asymptotics) Let C be a smooth projective curve with field of constants k of q elements, fix a closed point ∞ of C and let A be the coordinate ring of the affine curve C r {∞}. Then there exist ∆ ∈ N, ε> 0 such that m m h(Λ) =∆1∆Z(m)q + O (q − ε) Λ:RX(Λ)=m where the sum runs over all quadratic A-orders Λ and h(Λ) is the class number of Λ. 1 Affine buildings In this section we fix notations and cite results from other sources needed here. Main references are [2] and [5]. 1.1 The automorphism group Let X be a locally finite affine building. For the purpose of this paper, the most general definition of a building will do. So by a building we BUILDING LATTICES 4 understandapolysimplicialcomplexwhichistheunionofagiven family of affine Coxeter complexes, called apartments, such that any two chambers (=cells of maximal dimension, which is fixed) are contained in a common apartment and for any two apartments a, b containing chambers C, D there is a unique isomorphism a → b fixing C and D point-wise. A chamber is called thin if at every wall it has a unique neighbor chamber, it is called thick, if at each wall it neighbors at least two other chambers. The building is called thin or thick if all its chambers are. Note that our definition includes buildings which are not Bruhat-Tits. In higher dimensions, buildings tend to be of Bruhat-Tits type [7]. For buildings of dimension at most two the situation is drastically different. Indeed, Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building [3]. When speaking of “points” in X, we identify the complex X with its geometric realization. Note that the latter carries a topology as a CW- complex. In this topology, a set is compact if and only if it is closed and contained in a finite union of chambers. Note that an affine building is always contractible, see Section 14.4 of [20]. Definition 1.1.1. Generally,there are differentfamiliesofapartmentswhich make X a building, but there is a unique maximal family (Theorem 4.54 of [2]). In this paper, we will always choose the maximal family. Let Aut(X) be the automorphism group of the building X, that is, the set of all automorphisms g : X → X of the complex X which map apartments to apartments. In the geometric realisation these are cellular maps which are affine on each cell. Example 1.1.2. The Bruhat-Tits buliding of the group SL3(Q2) is a union of 2-dimensional apartments. These consist of chambers which in this case are equilateral triangles. BUILDING LATTICES 5 At each of these lines, one can switch to another apartment. We refer to a building with this kind of apartments as a building of type A2. Definition 1.1.3. (Aut(X) as a topological group.) The group Aut(X) will be equipped with the compact-open topology. This is the topology generated by all sets of the form L(K, U) = g ∈ Aut(X): g(K) ⊂ U , n o where K is any compact subset of X and U any open subset. In the case of a building, this topology can also be characterised by saying that a sequence (gn) in Aut(X) converges to g ∈ Aut(X) if and only if for every point x ∈ X there exists an integer n(x) such that for all n ≥ n(x) one has gnx = gx. Using this characterisation, it is easy to see that Aut(X) becomes a topological group, i.e., the composition Aut(X) × Aut(X) → Aut(X) is continuous. Lemma 1.1.4. The topological group Aut(X) is a locally compact group. This means that it is a Hausdorff space inwhich everypointhas acompactneighborhood. A basis of the unit-neighborhoods is given by the family of compact open subgroups KE = g ∈ Aut(X): ge = e ∀e∈E , n o where E ⊂ X is any finite set. BUILDING LATTICES 6 Proof. For the length of this proof we write G = Aut(X). Let g1, g2 ∈ G be two different elements. Then there exists x ∈ X with g1x , g2x. Then the sets U = {g ∈ G : gx = g1x} and V = {g ∈ G : gx = g2x} are open, satisfy U ∩ V = ∅ and g1 ∈ U, g2 ∈ V. So G is a Hausdorff space. We show that for E , ∅ the group KE is compact and open. Then for each g ∈ G the coset gKE is a compact neighborhood of g, so G is locally compact. To see that KE is open, let Ue be an open neighborhood of e for e ∈ E. Then the set of all g ∈ G with ge ∈ Ue is open. As every g ∈ G maps cells to cells and is an affine map on each cell, for each e ∈ E there exists an open neighborhood Ue of e such that ge ∈ Ue implies ge = e.