Lattices in complete rank 2 Kac–Moody groups
Let be a minimal Kac–Moody group of rank defined over the finite field , where with prime. Let be the topological Kac–Moody group obtained by completing . An example is , where is the field of formal Laurent series over . The group acts on its Bruhat–Tits building , a tree, with quotient a single edge. We construct new examples of cocompact lattices in , many of them edge-transitive. We then show that if cocompact lattices in do not contain –elements, the lattices we construct are the only edge-transitive lattices in , and that our constructions include the cocompact lattice of minimal covolume in . We also observe that, with an additional assumption on –elements in , the arguments of Lubotzky [L] for the case may be generalised to show that there is a positive lower bound on the covolumes of all lattices in , and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of .
A classical theorem of Siegel [Siegel45] states that the minimum covolume among lattices in is , and determines the lattice which realises this minimum. In the nonarchimedean setting, Lubotzky [L] determined the lattice of minimal covolume in , where is the field of formal Laurent series over .
The group has, in recent developments, been viewed as the first example of a complete Kac–Moody group of rank over the finite field . By definition, a complete Kac–Moody group is the completion of a minimal Kac–Moody group over a finite field, with respect to some topology. We use the completion in the “building topology”, as discussed in, for example, [CR]. Complete Kac–Moody groups are locally compact, totally disconnected topological groups, which may be thought of as infinite-dimensional analogues of semisimple algebraic groups (see Section LABEL:s:kac-moody below for details).
1.1. Constructions of cocompact lattices
Our first main result, Theorem 1.1 below, constructs many new cocompact lattices in rank complete Kac–Moody groups . It is interesting that there exist any cocompact lattices in such groups , since for , Kac–Moody groups of rank do not admit cocompact lattices (with the possible exception of those whose root systems contain a subsystem of type — see [CM, Remark 4.4]). In rank , the previous examples of cocompact lattices in non-affine that are known to us are the free Schottky groups constructed by Carbone–Garland [CG, Section 11], some of the lattices constructed by Rémy–Ronan [RR, Section 4.B], which in rank are free products of finite cyclic groups, and some of the lattices obtained as centralisers of certain involutions by Gramlich–Horn–Mühlherr [GHM, Section 7.3].
As we recall in Section LABEL:s:kac-moody below, the Kac–Moody groups that we consider have Bruhat–Tits building a regular tree , and the action of on induces an edge of groups
where and are the standard parahoric subgroups of and is the standard Iwahori subgroup. The kernel of the –action on is the finite group , the centre of (see [CR]). Now let be a cocompact lattice in which acts transitively on the edges of . Then as we recall in Section LABEL:s:lattices, is the fundamental group of an edge of groups
with , and finite groups (see Section 2.2 for background on graphs of groups).
We now state Theorem 1.1, which constructs new cocompact lattices in , most of them edge-transitive. We discuss our assumption that has symmetric generalised Cartan matrix after the statement of Theorem 1.1. There are some exceptional constructions for small values of which we then record in Theorem 1.2. Apart from the affine case , there are no known linear representations of the groups in Theorems 1.1 and 1.2.
Our notation is as follows. We write for the cyclic group of order and for the symmetric group on letters. Since for a finite field and the root system there exist at most two corresponding finite groups of Lie type, namely and , to avoid complications we use Lie-theoretic notation and write in both cases. (We will discuss this ambiguity whenever necessary.) We denote by a fixed maximal split torus of with . Then , and is isomorphic to a quotient of (the particular quotient depending upon ). Finally, the parahoric subgroups and of admit Levi decompositions (see Section LABEL:s:rank_2 and, in particular, Proposition LABEL:p:completion below), and for we denote by a Levi complement of . The group factors as , where is normalised by , and we denote by a non-split torus of such that is as big as possible. We say that two edge-transitive cocompact lattices and in are isomorphic if for and the obvious diagram commutes.
Let be a complete Kac–Moody group of rank with symmetric generalised Cartan matrix , , defined over the finite field of order where is prime. Then in the following cases, the group admits edge-transitive cocompact lattices of each of the following isomorphism types.
If then where for , and , and is any subgroup of .
If is odd and , assume also that . Then where for , , and is a subgroup of with , .
If is odd and , let
and let be the unique subgroup of of index .
If and then:
where for , where is of order , and .
Moreover, if , then also where for , where and .
where for , with , , and .
If , then also where either for , with and , or the description of in (2) holds.
In all other cases, admits a cocompact lattice which acts on the tree inducing a graph of groups of the form\setkeys
where the finite groups , and will be defined in Section LABEL:ss:define_groups below.
Our exceptional constructions for small values of are as follows.
Let be as in Theorem 1.1 above. Then in the following cases, admits cocompact edge-transitive lattices of the following isomorphism types.
When is odd and :
If , where for , where , and ; and
If , where for , where , and .
When is odd and :
If or , where for , , , and where is cyclic.
If , where for , , and with , being cyclic, and one of the following holds:
If or , where for , , , and with being cyclic.
We prove Theorems 1.1 and 1.2 in Section LABEL:s:cocompact. By the general theory of tree lattices (see Section LABEL:s:lattices), each or appearing in Theorems 1.1 and 1.2 yields a cocompact lattice in the full automorphism group of the tree . Since is much larger than the Kac–Moody group , the key point in proving Theorems 1.1 and 1.2 is to show that each such or embeds into as a cocompact lattice.
For this, we develop an embedding criterion, Proposition LABEL:p:embedding in Section LABEL:s:embedding below, which may be applied to construct lattices in any closed locally compact group acting transitively on the edges of a locally finite biregular tree. Our embedding criterion generalises [L, Lemma 3.1], which was used in [L] to construct edge-transitive lattices in the affine case . We are able to provide somewhat simpler proofs even in that case by using Bridson and Haefliger’s covering theory for complexes of groups [BH] instead of Bass’ covering theory for graphs of groups [B], since the theory for complexes of groups has a less complicated notion of morphism.
Our construction of the lattice in Theorem 1.1 generalises the construction in Example (6.2) of Lubotzky–Weigel [LW] of a cocompact lattice in when . Example (6.2) of [LW] uses the embedding criterion [LW, Theorem 5.4], which relies upon Bass’ covering theory [B] and is specific to the affine case, while we apply our criterion Proposition LABEL:p:embedding instead.
In the affine case , an element of has order if and only if it is unipotent. Hence by Godement’s Criterion, no –element is contained in a cocompact lattice. Moreover, any unipotent element of is contained in a conjugate of the upper unitriangular subgroup of , which is an infinite group isomorphic to . On the other hand, Lubotzky [L] showed that no cocompact lattice of contains non-trivial –elements (analogous to the classical Godement’s cocompactness criterion). For general , as we explain in Section LABEL:s:motivation below, there are many –elements which cannot be contained in any cocompact lattice , since they belong to a conjugate of a canonical subgroup of (see Section LABEL:s:action_ends for the definition of ). We thus make the following conjectures in the general case.
Cocompact lattices in do not contain –elements.
Any –element in is contained in a conjugate of the subgroup of .
1.3. Classification of edge-transitive lattices
Assuming Conjecture 1, we are able to classify the edge-transitive lattices in up to isomorphism, as follows.
The question of classifying amalgams of finite groups is, in general, difficult. An –amalgam is an amalgamated free product where has index in and index in . To avoid trivial examples, such an amalgam is said to be faithful if , and have no common normal subgroup. A deep theorem of Goldschmidt [G] established that there are only 15 faithful –amalgams of finite groups, and classified such amalgams. Goldschmidt and Sims conjectured that when both and are prime, there are only finitely many faithful –amalgams of finite groups (see [BK, F, G]). This conjecture remains open, except for the case , which was established by Djoković–Miller [DM], and the work of Fan [F], who proved the conjecture when the edge group is a –group, with a prime distinct from both and .
On the other hand, Bass–Kulkarni [BK] showed that if either or is composite, there are infinitely many faithful –amalgams of finite groups. The constructions in [BK] may be viewed as giving infinitely many non-isomorphic edge-transitive lattices in the automorphism group of an –biregular tree. Thus if Conjecture 1 holds, there are in contrast only finitely many edge-transitive lattices in Kac–Moody groups as in Theorem 1.1. We note that, since the action of such on is not in general faithful, an amalgam may embed as an edge-transitive lattice in even though it is not faithful (but its kernel will be at most the finite group ).
We prove Theorem 1.3 in Section LABEL:s:classification, making careful use of classical results on the subgroups of , and of order coprime to and their actions on the projective line .
1.4. Covolumes of cocompact lattices
If Conjecture 1 holds, there are also important consequences for the covolumes of lattices in . We recall in Section LABEL:s:lattices below that the Haar measure on may be normalised so that the covolume of a cocompact lattice with quotient graph of groups having two vertex groups and is equal to . Using this normalisation, we obtain the following.
In other words, for large enough, among cocompact lattices in which do not contain –elements the minimal covolume is , and this minimum is achieved by one of the lattices in Theorem 1.1.
Theorem 1.4 generalises Theorem 2 of Lubotzky [L] and part of Proposition C of Lubotzky–Weigel [LW], which together determine the cocompact lattices of minimal covolume in the affine case .
We prove Theorem 1.4 in Section LABEL:s:covolumes_cocompact. The proof is very delicate and has many cases, involving consideration of how various subgroups of , and of order coprime to can act on .
1.5. Covolumes of all lattices
Finally, assuming the stronger Conjecture 2, we obtain a lower bound on the covolume of all lattices in . As we recall in Section LABEL:s:kac-moody, the minimal Kac–Moody group has twin buildings , and there are two completions into which embeds, with acting on . As shown independently in Carbone–Garland [CG] and Rémy [R], a negative maximal parabolic subgroup of is a non-cocompact lattice in . We have:
Theorem 1.5 generalises Theorem 1 of Lubotzky [L], which establishes that the lattice of minimal covolume in is the maximal parabolic subgroup , a non-cocompact lattice. More recently, Golsefidy [G] has shown that for many Chevalley groups , the lattice of minimal covolume in is the non-cocompact lattice . Theorem 1.5 and the results of [L] and [G] thus contrast with Siegel’s original result [Siegel45] that the lattice of minimal covolume in is cocompact.
The proof of Theorem 1.5 is similar to that of [L, Theorem 1], after replacing the term “unipotent element” with “–element”. We thus omit the proof.
We are grateful to Benson Farb, Richard Lyons, Chris Parker and Kevin Wortman for helpful conversations. We would also like to thank Pierre-Emmanuel Caprace, Bertrand Rémy, Ron Solomon and an anonymous referee of an earlier version of this paper for suggestions and comments that improved this article. This work was completed during the first author’s stay at the Institute for Advanced Study and she would like to gratefully acknowledge the wonderful year she spent there. The second author thanks the Mathematical Sciences Research Institute and the London Mathematical Society for travel support.
This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
In Section 2.1 we recall basic definitions and establish notation for graphs and trees. Section 2.2 sketches the theory of graphs of groups, including covering theory. In Section LABEL:s:lattices we recall some theory of tree lattices. Section LABEL:s:kac-moody then presents background on the Kac–Moody groups that we consider, and in Section LABEL:s:finite_groups we recall some classical theorems concerning the finite subgroups of , and .
2.1. Graphs and trees
Let be a connected graph with sets of vertices and of oriented edges. The initial and terminal vertices of are denoted by and respectively. The map is orientation reversal, with and for . Given a vertex , we denote by the set of edges
with initial vertex .
Let and be graphs. A morphism of graphs is a function taking vertices to vertices and edges to edges, such that for every edge , and for .
Let be a simplicial tree with vertex set and edge set . A group is said to act without inversions on if for all and all edges , if preserves the edge then fixes pointwise.
Proposition 2.1 (Serre, Proposition 19, Section I.4.3 [S]).
Let be a finite group acting without inversions on a simplicial tree . Then there is a vertex of which is fixed by .
Now equip with a length metric by declaring each edge of to have length . Given an edge of and an integer , we define to be the union of the closed edges in all of whose points are distance at most from a point in the closed edge . By abuse of notation, we then define the distance between edges and of to be if , and to be if is in but not .
A geodesic line in is an isometry , and a geodesic ray in is an isometry , , such that is a vertex of ; in this case we say that the ray begins at the vertex . We will often identify a geodesic line or ray with its image in . We say that two geodesic rays and in are equivalent if their intersection is infinite. The boundary of , which is the same thing as the set of ends of , is the collection of equivalence classes of geodesic rays. Given a geodesic ray , we say that an end is determined by if belongs to the equivalence class .
If is a group of isometries of which acts without inversions, then an element is either elliptic, meaning that fixes at least one vertex of , or hyperbolic, meaning that does not fix any vertex and acts as a translation along its axis, a geodesic line in (see, for example, [BH, Chapter II.6]). If is hyperbolic then generates an infinite cyclic subgroup of .
Let be a hyperbolic isometry of , with axis . Then has exactly two fixed points in , which we denote by and . One of these fixed points is repelling and the other is attracting.
2.2. Bass–Serre theory
A graph of groups over a connected graph consists of an assignment of vertex groups for each and edge groups for each , together with monomorphisms for each .
Any action of a group on a tree without inversions induces a graph of groups over the quotient graph . See for example [B] for the definitions of the fundamental group and the universal cover of a graph of groups , with respect to a basepoint . The universal cover is a tree, on which acts by isometries inducing a graph of groups isomorphic to .
In the special case that is a graph of groups over an underlying graph which is a single edge , we say that is an edge of groups. Suppose and . Write for the edge group , and for let be the vertex group . The fundamental group is then isomorphic to the free product with amalgamation , and the universal cover is an –biregular tree, where and .
We now adapt definitions from covering theory for complexes of groups (see [BH, Chapter III.]) to graphs of groups, and recall a necessary result from covering theory. For the precise relationship between the category of graphs of groups and the category of complexes of groups over –dimensional spaces, see [Th, Proposition 2.1].
Definition 1 (Morphism of graphs of groups).
Let and be graphs of groups, with monomorphisms from edge groups to vertex groups respectively for and for . Let be a morphism of graphs. A morphism of graphs of groups over is given by:
a homomorphism of groups, for every ; and
an element for each such that the following diagram commutes, where :