4 Higher-dimensional complexes and their lattices
These problems are motivated by the theory of lattices in semisimple Lie groups, as well as by known results for tree lattices and for lattices in products of trees. Many questions can be asked for these specific cases as well. There is a great richness of examples of polyhedral complexes of dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:. These include buildings, both classical and non-classical (see Brown MR0969123 and Ronan MR1005533; key non-classical examples include right-angled buildings, hyperbolic buildings, and Kac--Moody buildings). Another important example is the Davis--Moussong complex for a Coxeter group (see Moussong). In dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:, a
dvipng error! exitcode was 2 (signal 0), transscript follows:--complex is a polygonal complex
dvipng error! exitcode was 2 (signal 0), transscript follows:such that the link of each vertex of
dvipng error! exitcode was 2 (signal 0), transscript follows:is the graph
dvipng error! exitcode was 2 (signal 0), transscript follows:, and each
dvipng error! exitcode was 2 (signal 0), transscript follows:--cell of
dvipng error! exitcode was 2 (signal 0), transscript follows:is a regular
dvipng error! exitcode was 2 (signal 0), transscript follows:--gon. This includes
dvipng error! exitcode was 2 (signal 0), transscript follows:which are neither buildings nor Davis--Moussong complexes, with
dvipng error! exitcode was 2 (signal 0), transscript follows:for example the Petersen graph (see~\'Swi\c{a}tkowski MR1612350). We refer the reader to the survey of Farb--Hruska--Thomas FarbHruskaThomas for further background, motivation and examples.
Problem 4.1 (Comments) Classify polyhedral complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:(of a certain class). Does "local data" determine
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark In general there may be uncountably many non-isomorphic
dvipng error! exitcode was 2 (signal 0), transscript follows:--complexes (Ballmann--Brin MR1279883, Haglund MR1133493). In contrast, right-angled buildings are specified up to isomorphism by a right-angled Coxeter system
dvipng error! exitcode was 2 (signal 0), transscript follows:and a set of cardinalities
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:(see for example Haglund--Paulin MR1957268). Some Fuchsian buildings are determined by their "holonomy", a local condition, (see Haglund MR1985452), but otherwise hyperbolic buildings have not been classified (Gaboriau--Paulin MR1877215). \'Swi\c{a}tkowski MR1612350 classified those
dvipng error! exitcode was 2 (signal 0), transscript follows:which are CAT(0) trivalent polygonal complexes of Platonic symmetry, meaning that
dvipng error! exitcode was 2 (signal 0), transscript follows:acts transitively on flags (vertex, edge, face). A specific case of this problem is: if large balls in two complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are isomorphic, is
dvipng error! exitcode was 2 (signal 0), transscript follows:isomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Problem 4.2 (Comments) When is
dvipng error! exitcode was 2 (signal 0), transscript follows:(non)discrete?
Remark The lattice theory of
dvipng error! exitcode was 2 (signal 0), transscript follows:is only interesting if
dvipng error! exitcode was 2 (signal 0), transscript follows:is nondiscrete. When
dvipng error! exitcode was 2 (signal 0), transscript follows:is the Davis--Moussong complex for a Coxeter system, this problem was solved by Haglund--Paulin MR1668359. \'Swi\c{a}tkowski MR1612350 solved this problem for
dvipng error! exitcode was 2 (signal 0), transscript follows:a CAT(0) trivalent polygonal complex of Platonic symmetry, meaning that
dvipng error! exitcode was 2 (signal 0), transscript follows:acts transitively on flags (vertex, edge, face). Very little is known for other cases.
Problem 4.3 (Comments) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a locally finite polyhedral complex, other than a product of trees, such that
dvipng error! exitcode was 2 (signal 0), transscript follows:acts cocompactly. Study the existence of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark For trees, there are existence theorems for both uniform (Bass--Kulkarni MR1065928) and nonuniform (Bass--Carbone--Rosenberg MR1794898) lattices. Various non-arithmetic constructions of lattices for certain higher-dimensional
dvipng error! exitcode was 2 (signal 0), transscript follows:may be found in work of Ballmann--Brin MR1383216, Benakli MR1314577, Bourdon MR1445387,MR1756974, Cartwright--Mantero--Steger--Zappa MR1232965,MR1232966, Gaboriau--Paulin MR1877215, Thomas MR2253444,MR2292989, and others. A specific case of this problem is: for
dvipng error! exitcode was 2 (signal 0), transscript follows:-complexes, is there an
dvipng error! exitcode was 2 (signal 0), transscript follows:such that for some
dvipng error! exitcode was 2 (signal 0), transscript follows:lattices exist, and for other
dvipng error! exitcode was 2 (signal 0), transscript follows:they do not?
Problem 4.4 (Comments) Classify lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:up to conjugacy, or up to commensurability.
Remark There are often uncountably many commensurability classes of nonuniform tree lattices (Bass--Lubotzky MR1794898, Farb--Hruska MR2214456), and similarly for right-angled buildings (Thomas MR2253444). There are some Fuchsian buildings such that all uniform lattices are commensurable (Haglund MR2240922), as is the case for trees (Leighton MR0693362).
Problem 4.5 (Comments) Rigidity: to what extent does a lattice
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:determine
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark More precisely, strong (Mostow) rigidity is the following. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:be nonpositively curved polyhedral complexes, and let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:. Find conditions on the
dvipng error! exitcode was 2 (signal 0), transscript follows:which guarantee that any abstract group isomorphism
dvipng error! exitcode was 2 (signal 0), transscript follows:extends to an isomorphism
dvipng error! exitcode was 2 (signal 0), transscript follows:. Further, determine when any two copies of
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:are conjugate in
dvipng error! exitcode was 2 (signal 0), transscript follows:.
There are examples of products of trees, some lattices of which are strongly rigid, others are not. What about other nonpositively curved 2-complexes?
A harder, more general problem is quasi-isometric rigidity, that is, determining when any quasi-isometry of
dvipng error! exitcode was 2 (signal 0), transscript follows:is a bounded distance from an isometry (automorphism). Quasi-isometric rigidity was shown for Euclidean buildings by Kleiner--Leeb MR1608566, for Bourdon's building
dvipng error! exitcode was 2 (signal 0), transscript follows:by Bourdon--Pajot MR1789183, and for general Fuchsian buildings by Xie MR2170496.
Another rigidity phenomenon that might be investigated is super-rigidity \`a la Margulis, which asks when homomorphisms defined on a lattice
dvipng error! exitcode was 2 (signal 0), transscript follows:extend to the ambient group
dvipng error! exitcode was 2 (signal 0), transscript follows:. See Lubotzky--Mozes--Zimmer MR1303226, Monod MR2219304,MR2163901, and Shalom MR1767270.
Problem 4.6 (Comments) When does a normal subgroup theorem hold for lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark Margulis' normal subgroup theorem (see MR1090825) states that if
dvipng error! exitcode was 2 (signal 0), transscript follows:is a lattice in a higher-rank, center-free, semisimple Lie group
dvipng error! exitcode was 2 (signal 0), transscript follows:, then every nontrivial normal subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:has finite index. Burger--Mozes MR1839489 proved a normal subgroup theorem for lattices in products of trees, and Bader--Shalom MR2207022 for lattices in products of very general locally compact groups.
Problem 4.7 (Comments) [Farb] When are nonuniform lattices finitely generated?
Remark If
dvipng error! exitcode was 2 (signal 0), transscript follows:has Kazhdan's property (T) then all lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:are finitely generated. For example,
dvipng error! exitcode was 2 (signal 0), transscript follows:is finitely generated as a nonuniform lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:. In contrast, all nonuniform tree lattices are infinitely generated (Serre MR1954121). For
dvipng error! exitcode was 2 (signal 0), transscript follows:, it is known in some cases whether or not
dvipng error! exitcode was 2 (signal 0), transscript follows:has property (T): see for example Ballmann--\'Swi\c{a}tkowski MR1465598, and Dymara--Januszkiewicz MR1946553. Some nonuniform lattices for Bourdon's building
dvipng error! exitcode was 2 (signal 0), transscript follows:are not finitely generated (Farb--Hruska FarbHruska).
Problem 4.8 (Comments) [Farb, Mozes] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:be infinite, locally finite
dvipng error! exitcode was 2 (signal 0), transscript follows:--complexes, with
dvipng error! exitcode was 2 (signal 0), transscript follows:acting cocompactly for
dvipng error! exitcode was 2 (signal 0), transscript follows:. If
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are isomorphic as groups, is
dvipng error! exitcode was 2 (signal 0), transscript follows:isometric/simplicially isometric/homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark Assumptions on the
dvipng error! exitcode was 2 (signal 0), transscript follows:could be added, for example nonpositive curvature or acyclicity. For trees, if
dvipng error! exitcode was 2 (signal 0), transscript follows:is isomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:, then
dvipng error! exitcode was 2 (signal 0), transscript follows:is isometric to
dvipng error! exitcode was 2 (signal 0), transscript follows:(Bass--Lubotzky MR1794898), and if the commensurators of uniform lattices
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are isomorphic as groups then
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are isometric (for
dvipng error! exitcode was 2 (signal 0), transscript follows:regular and biregular trees; Lubotzky--Mozes--Zimmer MR1303226). This question is open for products of trees, and for all other higher-dimensional complexes.
Problem 4.9 (Comments) [Farb, Mozes] Given
dvipng error! exitcode was 2 (signal 0), transscript follows:, a graph
dvipng error! exitcode was 2 (signal 0), transscript follows:and a finite
dvipng error! exitcode was 2 (signal 0), transscript follows:--complex
dvipng error! exitcode was 2 (signal 0), transscript follows:with
dvipng error! exitcode was 2 (signal 0), transscript follows:, is there an algorithm to determine, or can it be decided whether,
dvipng error! exitcode was 2 (signal 0), transscript follows:is irreducible (for
dvipng error! exitcode was 2 (signal 0), transscript follows:a product), residually finite, simple, linear, and so on? Find necessary and sufficient conditions for these properties.
Remark A particular case of this problem is where
dvipng error! exitcode was 2 (signal 0), transscript follows:is the quotient of a product of trees by a torsion-free lattice
dvipng error! exitcode was 2 (signal 0), transscript follows:acting freely, so that
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:is complete bipartite. Rattaggi Rattaggi has examples where
dvipng error! exitcode was 2 (signal 0), transscript follows:is simple and
dvipng error! exitcode was 2 (signal 0), transscript follows:is finite or infinite, and sufficient conditions for
dvipng error! exitcode was 2 (signal 0), transscript follows:to be irreducible, residually finite and simple are known.
Problem 4.10 (Comments) [de Cornulier, Shalom] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the Bruhat--Tits building associated to a higher-rank algebraic group
dvipng error! exitcode was 2 (signal 0), transscript follows:over a function field. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:. Is
dvipng error! exitcode was 2 (signal 0), transscript follows:arithmetic? Is
dvipng error! exitcode was 2 (signal 0), transscript follows:residually finite (asked by Burger--Mozes, Wise)?
Remark Note that the algebraic group
dvipng error! exitcode was 2 (signal 0), transscript follows:is cocompact in
dvipng error! exitcode was 2 (signal 0), transscript follows:, but is not of finite index (Tits MR0470099).
Problem 4.11 (Comments) [Farb] Characterize lattices coming from algebraic groups.
Remark More precisely, let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the Bruhat--Tits building associated to an algebraic group
dvipng error! exitcode was 2 (signal 0), transscript follows:over a local field. Characterize or recognize the lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:which are lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:. A possibility is finite generation of nonuniform lattices.
Problem 4.12 (Comments) Which lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:have dense commensurators in
dvipng error! exitcode was 2 (signal 0), transscript follows:? Is this equivalent to a lattice having infinite index in its commensurator?
Remark Margulis MR1090825 characterized arithmetic lattices among lattices in semisimple Lie groups as those with dense commensurators, and showed that this property is equivalent to a lattice having infinite index in its commensurator. Haglund MR1644077 has shown that for some Davis complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:, the commensurator of the associated Coxeter group (which may be regarded as a uniform lattice) is dense in
dvipng error! exitcode was 2 (signal 0), transscript follows:. Haglund Haglund07 and Barnhill--Thomas BarnhillThomas have shown that for
dvipng error! exitcode was 2 (signal 0), transscript follows:a right-angled building, the commensurator of the standard uniform lattice" is dense in
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 4.13 (Comments) For which
dvipng error! exitcode was 2 (signal 0), transscript follows:is the set of covolumes of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:bounded away from zero? discrete? For which
dvipng error! exitcode was 2 (signal 0), transscript follows:does
dvipng error! exitcode was 2 (signal 0), transscript follows:admit towers of uniform or nonuniform lattices?
Remark Kazhdan--Margulis MR0223487 showed that for
dvipng error! exitcode was 2 (signal 0), transscript follows:a noncompact simple real Lie group, such as
dvipng error! exitcode was 2 (signal 0), transscript follows:, the set of covolumes of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:has positive lower bound, and Wang MR0414787 proved that for such groups
dvipng error! exitcode was 2 (signal 0), transscript follows:, with the exceptions of
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:, the set of covolumes is discrete. The strong finiteness theorem of Borel--Prasad MR1019963 implies the same results for covolumes of lattices in higher-rank nonarchimedean Lie groups. Dramatically different results hold for tree lattices: see work of Bass--Kulkarni MR1065928, Bass--Lubotzky MR1794898, Rosenberg Rosenberg and Farb--Hruska MR2214456. Thomas has shown that covolumes of lattices for right-angled buildings are very similar to those for trees MR2253444, and that for some Fuchsian buildings MR2292989 and Davis complexes Thomas07, the set of covolumes is not discrete.
A tower of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:is an infinite strictly ascending sequence of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:. If
dvipng error! exitcode was 2 (signal 0), transscript follows:admits a tower of lattices then the set of covolumes of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:does not have positive lower bound. Towers of uniform tree lattices were constructed by Bass--Kulkarni MR1065928 and Rosenberg Rosenberg, and nonuniform towers by Carbone--Rosenberg MR1849263. Thomas has shown that for
dvipng error! exitcode was 2 (signal 0), transscript follows:a right-angled building,
dvipng error! exitcode was 2 (signal 0), transscript follows:admits towers of uniform and nonuniform lattices MR2253444.