skip to main content


Title: Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups
We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist.  more » « less
Award ID(s):
1764421
NSF-PAR ID:
10336384
Author(s) / Creator(s):
;
Date Published:
Journal Name:
ArXivorg
ISSN:
2331-8422
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and degree of polynomial growth both bounded effectively.The third and fourth authors proved the optimal bound on the degree of polynomial growth of this subgroup, at the expense of making some other parts of the result ineffective.In the present paper, we prove the optimal bound on the degree of polynomial growth without making any losses elsewhere.As a consequence, we show that there exist explicit positive numbers ε d \varepsilon_{d} such that, in any group with growth at least a polynomial of degree 𝑑, the growth is at least ε d ⁢ n d \varepsilon_{d}n^{d} .We indicate some applications in probability; in particular, we show that the gap at 1 for the critical probability for Bernoulli site percolation on a Cayley graph, recently proven to exist by Panagiotis and Severo, is at least exp ⁡ { - exp ⁡ { 17 ⁢ exp ⁡ { 100 ⋅ 8 100 } } } \exp\{-\exp\{17\exp\{100\cdot 8^{100}\}\}\} . 
    more » « less
  2. We show that if a finitely generated group $G$ has a nonelementary WPD action on a hyperbolic metric space $X$ , then the number of $G$ -conjugacy classes of $X$ -loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$ . As an application we prove that for $N\geq 3$ the number of distinct $\text{Out}(F_{N})$ -conjugacy classes of fully irreducible elements $\unicode[STIX]{x1D719}$ from an $R$ -ball in the Cayley graph of $\text{Out}(F_{N})$ with $\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$ of the order of $R$ grows exponentially in $R$ . 
    more » « less
  3. We study the category of S p \mathbf {Sp} -equivariant modules over the infinite variable polynomial ring, where S p \mathbf {Sp} denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M M fits into an exact triangle T → M → F → T \to M \to F \to where T T is a finite length complex of torsion modules and F F is a finite length complex of “free” modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras Sym ⁡ ( C ∞ ⊕ ⋀ 2 C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \bigwedge ^2{\mathbf {C}^{\infty }}) and Sym ⁡ ( C ∞ ⊕ Sym 2 ⁡ C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \operatorname {Sym}^2{\mathbf {C}^{\infty }}) are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian. 
    more » « less
  4. Abstract This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces. 
    more » « less
  5. Abstract

    The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field and an extension ω of ν to a finite extensionLofK. Then we study when the valuation ring of ω is essentially finitely generated over the valuation ring of ν. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

     
    more » « less