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<n-1}, {\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})}.

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