** Higher dimensional Dehn functions for some CAT(0) groups **

**1. Higher dimensional Dehn functions **

** 1.1. Filling **

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 1Let where is a symmetric space, Euclidean space, Euclidean building. Let act geometrically on . Let be the Euclidean rank of . Then1. 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 .