## Notes of Enrico Leuzinger’s lecture

Higher dimensional Dehn functions for some CAT(0) groups

1. Higher dimensional Dehn functions

1.1. Filling

Let ${M}$ be a ${k}$-connected Riemannian manifold with bounded geometry. The ${k}$-th (homotopical) filling function is defned as follows. For every smooth ${k}$-sphere ${f:S^k \rightarrow M}$, the filling volume is the ${\inf}$ of volumes of extensions of ${f}$ to the ${k+1}$-ball. Then set

$\displaystyle \begin{array}{rcl} Fill^{k+1}_M(L)=\sup\textrm{ of filling volumes of all }k\textrm{-spheres of volume } \leq L. \end{array}$

The growth type of ${Fill^{k+1}_M}$ is a quasi-isometry invariant.

1.2. Combinatorial analogue

Let ${\Gamma}$ be a group of finiteness type ${F_{k+1}}$, i.e. it has a classifying space with finite ${k+1}$-skeleton. So there exists a ${k}$-connected ${k+1}$-complex ${K}$ on which ${\Gamma}$ acts cocompactly. Then combinatorial ${k}$-spheres can be filled, so filling volume make sense, and so does ${Fill^{k+1}_K}$, which is called the ${k}$-dimensional Dehn function.

Its growth type is a quasi-isometry invariant (Alonso). Therefore we view it as an invariant of ${\Gamma}$, ${ \delta^k_\Gamma}$. If ${\Gamma}$ is quasi-isometric to ${M}$ which is ${k}$-connected and has bounded geometry, then ${\delta^k_\Gamma}$ is equivalent to ${Fill^{k+1}_M}$.

1.3. Classical Dehn function

${k=1}$ yields the classical Dehn function, which has been thoroughly studied.

Gromov: a group ${\Gamma}$ is hyperbolic ${\Leftrightarrow}$ ${\delta^1_\Gamma}$ is linear.

Example 1 ${Sl(2,Z)}$.

Gromov: nothing between linear and quadratic.

Many groups have quadratic Dehn function: ${{\mathbb Z}^n}$, ${Sl(n,{\mathbb Z})}$ for ${n>4}$ (Young, still open for ${n=4}$), mapping class groups.

Many groups have exponential Dehn function: lattices in ${Sol}$, ${Sl(3,{\mathbb Z})}$, ${Out(F_n)}$ for ${n>2}$ (Bridson).

Beyond quadratic, nearly every growth type is achievable.

2. Survey of ${k>1}$

${k>1}$ 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 ${X=S \times E \times B}$ where ${S}$ is a symmetric space, ${E}$ Euclidean space, ${B}$ Euclidean building. Let ${\Gamma}$ act geometrically on ${X}$. Let ${r}$ be the Euclidean rank of ${X}$. Then

1. for ${k \leq r}$, ${\delta^k_\Gamma (L) \sim L^{k+1/k}}$.

2. for ${k>r}$, ${\delta^k_\Gamma (L) \sim L}$.

Wenger showed that all ${CAT(0)}$ spaces satisfy the Euclidean profile, this proves (1). (2) results from a cone construction.

3. A conjecture

Uniform ${S}$-arithmetic groups are examples of such lattices. I am interested in the non-uniform case. Let ${K}$ be a global field (number field or function field). Let ${S}$ be a finite set of valuations on ${K}$ including all archimedean ones. Let ${O_S}$ denote the group of integers (elements of ${K}$ whose valuations are ${\leq 1}$ except for ${S}$-valuations). Let ${G}$ be a reductive matrix group defined over ${K}$, and ${\Gamma=G(O_S)}$ the matrices in ${G}$ with entries in ${O_S}$.

If ${G}$ is ${K}$-anisotropic, ${\Gamma}$ is uniform. Examples exist for all ${G}$ if ${char(K)=0}$ (Borel-Harder). If ${char(K)>0}$, they exist only if ${G}$ is of type ${A}$ (Harder).

Conjecture. Let ${G}$ be isotropic. Then ${\Gamma}$ is a non-uniform lattice in a space ${X = S \times E \times B}$. Let ${r=rank(X)}$ and ${n=dim(X)}$.

1. If ${k < rank(X)-1}$, ${\delta^k_\Gamma(L) \sim L^{k+1/k}}$.

If ${char(K)>0}$, ${\Gamma}$ is of type ${F_{r-1}}$ but not ${F_r}$, so ${\delta^r-1_\Gamma}$ is not defined. So from now on, ${char(K)=0}$.

2. If ${k=r-1}$, ${\delta^k_\Gamma(L)}$ is exponential.

3. If ${r-1 \leq k, ${\delta^k_\Gamma(L)}$ is superlinear.

4. If ${k \geq n-1}$, ${\delta^k_\Gamma(L) \sim L}$.

(4) is easy (non amenability for ${k=n-1}$). (1) is known only for certain ${Sl(n,{\mathbb Z})}$.