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Register a new userAlexandrov spaces with large volume growth
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Authors
Michael Munn
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Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,mathcal{H}^n)$ has finite topological type.
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We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karchers classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.
The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.
We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given $N, D$, there exists $epsilon(N, D)>0$, such that for $epsilon<epsilon(N, D)$, if $X$ is an $N$-dimensional Alexandrov space with curvature $geq -1$, $operatorname{diam}(X)leq D, h(X)geq N-1-epsilon$, then $X$ is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity of cite{CRX} to Alexandrov spaces.
Let $G$ be a finite group with symmetric generating set $S$, and let $c = max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley graph of $G$ is bounded by a function of only $c$ and $k$. More specifically, the multiplicity is at most $exp((log c)(log c + log k))$. Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with non-negative Ricci curvature, then the multiplicity of the $k$th eigenvalue of the Laplace-Beltrami operator on $X$ is at most $exp(n^2 + n log k)$. The first result (for $k=2$) yields the following group-theoretic application. There exists a normal subgroup $N$ of $G$, with $[G : N] leq alpha(c)$, and such that $N$ admits a homomorphism onto the cyclic group $Z_M$, where $M geq |G|^{delta(c)}$ and $alpha(c), delta(c) > 0$ are explicit functions depending only on $c$. This is the finitary analog of a theorem of Gromov which states that every infinite group of polynomial growth has a subgroup of finite index which admits a homomorphism onto the integers. This addresses a question of Trevisan, and is proved by scaling down Kleiners proof of Gromovs theorem. In particular, we replace the space of harmonic functions of fixed polynomial growth by the second eigenspace of the Laplacian on the Cayley graph of $G$.
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