S. M. Gersten, D. F. Holt, T. R. Riley
We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c+1 and a linear upper bound on its filling length function.
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We show that arithmetic subgroups of semisimple groups of relative Q-type A_n, B_n, C_n, D_n, E_6, or E_7 have an exponential lower bound to their isoperimetric inequality in the dimension that is 1 less than the real rank of the semisimple group.
Homological Isoperimetric Inequalities for Kernels of Free Extensions of Type $FP_2$
本文通过定义同调面积-半径对，利用表面图示适应Gersten和Short的证明，得到关于$FP_2$类型子群的同调等周不等式。
It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. For each positive rational number $s$ we construct pairs of finitely presented groups $H\subset G$ such that the distortion of $H$ in $G$ is $\simeq n^s$. And for each $s\ge 1$ we also construct finitely presented groups whose isodiametric function is $\simeq n^s$.
We show that the mapping class group of a handlebody of genus at least 2 has a Dehn function of at most exponential growth type.
Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl^α$ for $α>2$ then it satisfies the averaged isoperimetric inequality $δ^{\text{avg}}(l)<C'l^{α/2}$. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.
We show that every oriented $n$-dimensional Poincaré duality group over a $*$-ring $R$ is amenable or satisfies a linear homological isoperimetric inequality in dimension $n-1$. As an application, we prove the Tits alternative for such groups when $n=2$. We then deduce a new proof of the fact that when $n=2$ and $R = \mathbb Z$ then the group in question is a surface group.
We prove an inequality, valid on any finitely generated group with a fixed finite symmetric generating set, involving the growth of successive balls, and the average length of an element in a ball. It generalizes recent improvements of the Coulhon Saloff-Coste inequality. We reformulate the inequality in terms of the Følner function; in the case the finitely generated group is amenable with exponential growth, this allows us to express the best possible (outer) constant in the Coulhon Saloff-Coste isoperimetric inequality with the help of a formula involving the growth rate and the asymptotic …
We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize existence of primitives of the volume form with pre…
We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most …
Isoperimetric inequalities in Hadamard spaces of asymptotic rank two
针对渐近秩为二的Hadamard空间，证明了维数至少为2的一般循环具有线性等周填充不等式，推广了Druţu-Lang-Papasoglu-Stadler关于Lipschitz 2-球的同伦论结果。
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite…
Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every $d$ there are unbounded degree simplicial complexes of dimension $d$ with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders exist for $d \geq 2$. We present an explicit construction of bounded degree complexes of dimension $d=2$ which are topological…
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