Continuity of asymptotic entropy on wreath products

We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products $ A \wr B = \bigoplus_B A \rtimes B $, where $A$ is any countable group and $B$ is a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. As one step in proving the above, we show that on any countable group $G$ the probability that the $μ$-random walk on $G$ never returns to the identity is continuous in $μ$, for measures $μ$ such that the semigroup generated by the support of $μ$ contains a finitely generated subgroup of at least cubic growth. Finally, we show that among random walks on a group $G$ that admit a separable completely metrizable space $X$ as a model for their Poisson boundary, the weak continuity of the associated harmonic measures on $X$ implies the continuity of the asymptotic entropy. This result recovers the continuity of asymptotic entropy on known cases, such as Gromov hyperbolic groups and acylindrically hyperbolic groups, and extends it to new classes of groups, including linear groups and groups acting on $\mathrm{CAT}(0)$ spaces.

💡 Research Summary

The paper addresses the continuity of asymptotic entropy h(μ) as a function of the step distribution μ for random walks on countable groups. While the entropy criterion (Kaimanovich–Vershik) links finite Shannon entropy H(μ) to non‑Liouville behavior, it remained unclear whether h(μ) varies continuously under perturbations of μ, especially for infinitely supported or non‑symmetric measures. The authors resolve this problem for a broad class of wreath products and, more generally, for groups whose Poisson boundary can be modeled by a Polish space.

Main results

1. Continuity on wreath products (Theorem 1.2).
   Let A be any countable group and B a countable hyper‑FC‑central group that contains a finitely generated subgroup of at least cubic growth (e.g. ℤ³ or the integer Heisenberg group). For any non‑degenerate probability measure μ on the wreath product A ≀ B with finite Shannon entropy, if a sequence μ_k converges pointwise to μ and H(μ_k)→H(μ), then h(μ_k)→h(μ). The proof exploits the relation between the entropy of the walk on A ≀ B and the number of distinct sites visited by its projection onto B. This number is controlled by the escape probability of the induced walk on B.

2. Continuity of the escape probability (Theorem 1.4).
   For a countable group G, assume the semigroup generated by supp(μ) contains a finitely generated subgroup of at least cubic growth. Then the probability that a μ‑random walk never returns to the identity, p_esc(μ), depends continuously on μ: pointwise convergence of μ_k to μ implies p_esc(μ_k)→p_esc(μ). The argument uses a uniform version of a comparison lemma of Coulhon and Saloff‑Coste and Varopoulos’ classification of groups admitting recurrent non‑degenerate walks (which must have at most quadratic growth). Hence discontinuities of p_esc can only arise in groups of quadratic or slower growth.

3. From harmonic measure continuity to entropy continuity (Theorem 1.5).
   Let G be countable, μ and μ_k non‑degenerate with finite entropy, and suppose there exists a Polish space X on which G acts continuously. If (X,ν) and (X,ν_k) are respectively the Poisson boundaries of (G,μ) and (G,μ_k) and ν_k converges weakly to ν, then h(μ_k)→h(μ). The proof combines three ingredients:

   • Upper semi‑continuity of asymptotic entropy (Avez–Vershik).
   • The Kaimanovich–Vershik representation of h(μ) as the average Kullback–Leibler divergence between ν and its translates.
   • Lower semi‑continuity of Kullback–Leibler divergence under weak convergence of probability measures on Polish spaces (Posner).

4. Corollaries for concrete classes (Corollary 1.6).
   The above theorem yields continuity for:

   • Acylindrically hyperbolic groups (previously known but recovered here).
   • Zariski‑dense discrete subgroups of SL_d(ℝ) with finite logarithmic moment.
   • Proper, cocompact actions on proper CAT(0) spaces with rank‑one elements and finite first moment.
   • Proper, cocompact actions on finite‑dimensional CAT(0) cube complexes with finite entropy and logarithmic moment.

These include new cases such as SL_d(ℤ) for d≥3 and lattices in higher‑rank symmetric spaces, where continuity of h(μ) was not previously established even for measures supported on a fixed finite generating set.

Methodological insights

• The paper shows that the “escape probability” serves as a bridge between geometric growth of the base group and entropy continuity on the wreath product.
• By formulating entropy continuity in terms of weak convergence of harmonic measures on a concrete boundary model, the authors avoid delicate combinatorial estimates and obtain a clean, abstract criterion.
• The growth condition (at least cubic) is shown to be optimal: groups with at most quadratic growth admit recurrent non‑degenerate walks, leading to possible discontinuities of p_esc and consequently of h(μ).

Open problems

The authors pose the continuity question for infinite Burnside groups B(m,n) (with large exponent n) where the base group has exponential growth but no known hyper‑FC‑central structure. This suggests that further structural conditions beyond growth may be required.

Conclusion

The work unifies several strands—random walk transience, Poisson boundary theory, and information‑theoretic entropy—into a coherent framework that yields continuity of asymptotic entropy for a wide array of groups, notably wreath products with sufficiently fast‑growing bases and many geometrically significant groups (hyperbolic, acylindrically hyperbolic, linear, CAT(0)). It both recovers known results and opens new directions for studying entropy regularity in groups where the Poisson boundary is non‑trivial or not yet explicitly described.