Higher dimensional Dehn functions for some CAT(0) groups
1. Higher dimensional Dehn functions
Let be a -connected Riemannian manifold with bounded geometry. The -th (homotopical) filling function is defned as follows. For every smooth -sphere , the filling volume is the of volumes of extensions of to the -ball. Then set
The growth type of is a quasi-isometry invariant.
1.2. Combinatorial analogue
Let be a group of finiteness type , i.e. it has a classifying space with finite -skeleton. So there exists a -connected -complex on which acts cocompactly. Then combinatorial -spheres can be filled, so filling volume make sense, and so does , which is called the -dimensional Dehn function.
Its growth type is a quasi-isometry invariant (Alonso). Therefore we view it as an invariant of , . If is quasi-isometric to which is -connected and has bounded geometry, then is equivalent to .
1.3. Classical Dehn function
yields the classical Dehn function, which has been thoroughly studied.
Gromov: a group is hyperbolic is linear.
Example 1 .
Gromov: nothing between linear and quadratic.
Many groups have quadratic Dehn function: , for (Young, still open for ), mapping class groups.
Many groups have exponential Dehn function: lattices in , , for (Bridson).
Beyond quadratic, nearly every growth type is achievable.
2. Survey of
has been little studied. Here is the list of groups for which all higher Dehn functions are known.
– Hyperbolic groups : all higher Dehn functions are linear (Mineyev, Lang).
– Heisenberg groups : all are polynomial (Young).
– Following theorem.
Theorem 1 Let where is a symmetric space, Euclidean space, Euclidean building. Let act geometrically on . Let be the Euclidean rank of . Then
1. for , .
2. for , .
Wenger showed that all spaces satisfy the Euclidean profile, this proves (1). (2) results from a cone construction.
3. A conjecture
Uniform -arithmetic groups are examples of such lattices. I am interested in the non-uniform case. Let be a global field (number field or function field). Let be a finite set of valuations on including all archimedean ones. Let denote the group of integers (elements of whose valuations are except for -valuations). Let be a reductive matrix group defined over , and the matrices in with entries in .
If is -anisotropic, is uniform. Examples exist for all if (Borel-Harder). If , they exist only if is of type (Harder).
Conjecture. Let be isotropic. Then is a non-uniform lattice in a space . Let and .
1. If , .
If , is of type but not , so is not defined. So from now on, .
2. If , is exponential.
3. If , is superlinear.
4. If , .
(4) is easy (non amenability for ). (1) is known only for certain .