[Paper Review] Embeddings of discrete groups and the speed of random walks

Assaf Naor, Yuval Peres|ArXiv.org|Aug 7, 2007

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes a sharp upper bound on the equivariant compression exponent of discrete groups into Banach spaces with power-type smoothness, linking it to the speed of random walks on the group. It proves that for a Banach space $X$ with modulus of smoothness of power type $p$, the equivariant Hilbert compression exponent satisfies $\alpha^{\#}_{X}(G) \leq \frac{1}{p\beta^{*}(G)}$, where $\beta^{*}(G)$ measures the growth rate of the expected distance in simple random walks. This resolves a question of Tessera and generalizes results of Guentner and Kaminker.

ABSTRACT

For a finitely generated group G and a banach space X let α^*_X(G) (respectively α^#_X(G)) be the supremum over all α\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G o X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α^*(G)=α^*_{L_2}(G) (respectively α^#(G)= α_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then α^#_X(G)\le \frac{1}{pβ^*(G)}. Here β^*(G) is the largest β\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^β, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if α^*(G)\ge 1/2 then α^*(G\bwr \Z)\ge \frac{2α^*(G)}{2α^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then α^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type.

Motivation & Objective

• To establish a sharp upper bound on the equivariant compression exponent of finitely generated groups into Banach spaces with power-type smoothness.
• To connect the speed of random walks on a group to its geometric embedding properties in Banach spaces.
• To resolve open questions regarding the Hilbert compression exponent of lamplighter groups and the relationship between Markov type and Enflo type.
• To investigate the exact value of the Hilbert compression exponent for iterated lamplighter groups $\mathbb{Z}_{(k)} = \mathbb{Z}_{(k-1)} \wr \mathbb{Z}$.

Proposed method

• Define $\alpha^{\#}_{X}(G)$ as the supremum of $\alpha \geq 0$ for which there exists an equivariant Lipschitz embedding $f: G \to X$ with $\|f(x)-f(y)\| \geq c \cdot d_G(x,y)^\alpha$.
• Introduce $\beta^{*}(G)$ as the largest $\beta \geq 0$ such that the expected distance $\mathbb{E}[d_G(W_t,e)]$ grows at least as $ct^\beta$ for simple random walks $\{W_t\}$ on the Cayley graph of $G$.
• Use the modulus of smoothness of power type $p$ in the target Banach space $X$ to derive the inequality $\alpha^{\#}_{X}(G) \leq \frac{1}{p\beta^{*}(G)}$, generalizing prior results.
• Apply the bound to $L_p$ spaces, showing $\alpha^{\#}_p(G) \leq \frac{1}{p\beta^{*}(G)}$, and prove sharpness in the case $X = L_p$.
• Use the bound to derive the exact value of the Hilbert compression exponent for iterated lamplighter groups: $\alpha^{*}(\mathbb{Z}_{(k)}) = \frac{1}{2 - 2^{1-k}}$.
• Use the result to show that cyclic lamplighter groups $C_2 \wr C_n$ embed into $L_1$ with uniformly bounded distortion, answering a question of Lee, Naor, and Peres.

Experimental results

Research questions

• RQ1 Does the inequality $\alpha^{\#}_{X}(G) \leq \frac{1}{p\beta^{*}(G)}$ hold for all finitely generated groups $G$ and Banach spaces $X$ with modulus of smoothness of power type $p$?
• RQ2 Is it true that $\alpha^{*}(G \wr \mathbb{Z}) = \frac{2\alpha^{*}(G)}{2\alpha^{*}(G) + 1}$ for all finitely generated amenable groups $G$?
• RQ3 Does every finitely generated amenable group $G$ satisfy $\alpha^{*}(G) \geq \frac{1}{2}$?
• RQ4 Can edge Markov type 2 imply Enflo type $p$ for some $p > 1$?
• RQ5 Is there a finitely generated group with edge Markov type 2 but no Enflo type $p$ for $p > 1$?

Key findings

• The paper establishes the sharp bound $\alpha^{\#}_{X}(G) \leq \frac{1}{p\beta^{*}(G)}$ for any Banach space $X$ with modulus of smoothness of power type $p$, generalizing a result of Guentner and Kaminker.
• For $X = L_p$, the bound $\alpha^{\#}_p(G) \leq \frac{1}{p\beta^{*}(G)}$ is sharp, and equality is achieved in certain cases.
• The Hilbert compression exponent of the iterated lamplighter group $\mathbb{Z}_{(k)} = \mathbb{Z}_{(k-1)} \wr \mathbb{Z}$ is exactly $\alpha^{*}(\mathbb{Z}_{(k)}) = \frac{1}{2 - 2^{1-k}}$, resolving a question of Tessera.
• The cyclic lamplighter groups $C_2 \wr C_n$ embed into $L_1$ with uniformly bounded distortion, answering a question posed by Lee, Naor, and Peres.
• The result implies that edge Markov type 2 does not imply Enflo type $p$ for any $p > 1$, providing a counterexample to a natural conjecture.
• For $1 < p < 2$, the paper gives bounds on $\alpha^{*}_p(\mathbb{Z} \wr \mathbb{Z})$, showing $\frac{p}{2p-1} \leq \alpha^{*}_p(\mathbb{Z} \wr \mathbb{Z}) \leq \min\left\{\frac{p+1}{2p}, \frac{4}{3p}\right\}$.