2 Tree lattices
Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a locally finite tree. The standard reference for lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:is Bass--Lubotzky MR1794898; see also Lubotzky's survey MR1320284. For uniform tree lattices many questions have been answered, but there are still some significant open problems. Much less is known about nonuniform tree lattices. For instance, there is no classification.
The main analogy, as explained by Lubotzky in MR1320284, is with lattices in rank one Lie groups over nonarchimedean local fields, such as
dvipng error! exitcode was 2 (signal 0), transscript follows:, since the Bruhat--Tits buildings of such groups are regular or biregular trees. More recently, lattices have been constructed in rank 2 Kac--Moody groups over finite fields, which also have trees as their buildings (see Carbone--Garland MR2017720). Note that the definition of rank is different for Kac--Moody groups: in this context, higher rank means rank
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 2.1 (Comments) [Farb] There is the following analogy:
dvipng error! exitcode was 2 (signal 0), transscript follows:is to
dvipng error! exitcode was 2 (signal 0), transscript follows:as
dvipng error! exitcode was 2 (signal 0), transscript follows:is to the group of quasi-conformal homeomorphisms
dvipng error! exitcode was 2 (signal 0), transscript follows:. What are the maximal closed subgroups of
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:? Can we use this analogy to somehow measure the distance between lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:, as in Teichm\"uller space?
Remark This analogy is motivated by, for example, the fact that two non-commensurable uniform lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:are commensurable in
dvipng error! exitcode was 2 (signal 0), transscript follows:, and similarly with
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 2.2 (Comments) [Lubotzky--Mozes] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a uniform lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:, where
dvipng error! exitcode was 2 (signal 0), transscript follows:is the
dvipng error! exitcode was 2 (signal 0), transscript follows:--regular tree. Is the commensurator
dvipng error! exitcode was 2 (signal 0), transscript follows:simple?
Remark
dvipng error! exitcode was 2 (signal 0), transscript follows:is the subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:generated by vertex stabilizers, which is here the group of type-preserving automorphisms. Recall that two subgroups
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:are commensurable if
dvipng error! exitcode was 2 (signal 0), transscript follows:has finite index in both
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. The commensurator of
dvipng error! exitcode was 2 (signal 0), transscript follows:is the group
dvipng error! exitcode was 2 (signal 0), transscript follows:
The group
dvipng error! exitcode was 2 (signal 0), transscript follows:is well-defined up to conjugacy, since all uniform lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:are, up to conjugacy, commensurable (Leighton MR0693362). This problem asks if
dvipng error! exitcode was 2 (signal 0), transscript follows:is simple as a group (that is, has no nontrivial normal subgroups).
The analogy is with the commensurator of an arithmetic lattice being a simple group.
This problem has been open for some time. The answer is probably similar for a biregular tree. One could ask: when is the commensurator of a nonuniform tree lattice simple?
Problem 2.3 (Comments) [Lubotzky--Mozes] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:be uniform lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:. If
dvipng error! exitcode was 2 (signal 0), transscript follows:(where we really mean equal as groups, not just conjugate), is
dvipng error! exitcode was 2 (signal 0), transscript follows:commensurable to
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark Recall that two subgroups
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:are commensurable if
dvipng error! exitcode was 2 (signal 0), transscript follows:has finite index in both
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. The commensurator of
dvipng error! exitcode was 2 (signal 0), transscript follows:is the group
dvipng error! exitcode was 2 (signal 0), transscript follows:
Let
dvipng error! exitcode was 2 (signal 0), transscript follows:. The group
dvipng error! exitcode was 2 (signal 0), transscript follows:is well-defined up to conjugacy, since all uniform lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:are, up to conjugacy, commensurable (Leighton, MR0693362). This is then the same question as: is the normalizer
dvipng error! exitcode was 2 (signal 0), transscript follows:? The answer is yes for algebraic groups. The answer is probably similar for a biregular tree. A further generalization would be to uniform trees
dvipng error! exitcode was 2 (signal 0), transscript follows:, that is, those where
dvipng error! exitcode was 2 (signal 0), transscript follows:admits a uniform lattice (characterized by Bass--Kulkarni MR1065928).
Problem 2.4 (Comments) For
dvipng error! exitcode was 2 (signal 0), transscript follows:a nonuniform lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:, what are the possible closures of the commensurator
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:? Find an example where the commensurator is neither dense nor discrete. Give a criterion to ensure density or discreteness of the commensurator.
Remark Recall that two subgroups
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:are commensurable if
dvipng error! exitcode was 2 (signal 0), transscript follows:has finite index in both
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. Examples due to Bass, Lubotzky and Mozes are known with dense commensurator and with discrete commensurator (see MR1794898), and density of the commensurator was shown to hold for many nonuniform tree lattices of so-called Nagao type by Abramenko--R\'emy AbramenkoRemy. It is not known whether these are the only possibilities. Arithmetic lattices in connected semisimple Lie groups (with trivial center and no compact factors) are characterized by having dense commensurators (Margulis MR0492072). All uniform tree lattices are known to have dense commensurators (Liu MR1273278).
Problem 2.5 (Comments) [Mozes] The group
dvipng error! exitcode was 2 (signal 0), transscript follows:has Bruhat--Tits building the
dvipng error! exitcode was 2 (signal 0), transscript follows:--regular tree
dvipng error! exitcode was 2 (signal 0), transscript follows:. Prove that nonuniform lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:have dense commensurator in
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark The group
dvipng error! exitcode was 2 (signal 0), transscript follows:is of infinite index in
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 2.6 (Comments) [Goldschmidt--Sims] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the
dvipng error! exitcode was 2 (signal 0), transscript follows:--biregular tree. If
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are prime, does
dvipng error! exitcode was 2 (signal 0), transscript follows:admit only finitely many (conjugacy classes of) uniform lattices with quotient an edge?
Remark This question can be phrased in purely group-theoretic terms: are there only finitely many amalgams of finite groups
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:, and any normal subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:which remains normal under the inclusion into both
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:is trivial? When
dvipng error! exitcode was 2 (signal 0), transscript follows:, there are exactly 15 such amalgams (Goldschmidt MR0569075). Djokovic--Miller MR0586434 showed that there are exactly 7 such amalgams when
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. Fan MR0826134 showed that there are only finitely many such amalgams
dvipng error! exitcode was 2 (signal 0), transscript follows:when the order of
dvipng error! exitcode was 2 (signal 0), transscript follows:is a power of a prime distinct from the primes
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. If either
dvipng error! exitcode was 2 (signal 0), transscript follows:or
dvipng error! exitcode was 2 (signal 0), transscript follows:is not prime, there are infinitely many such amalgams, as shown by Bass--Kulkarni MR1065928. In higher dimensions, Glasner MR2015283 showed that for the product of trees
dvipng error! exitcode was 2 (signal 0), transscript follows:, when
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are both prime there are (up to conjugacy) only finitely many irreducible lattices with quotient a square.