Viktor Losert
We show that a compactly generated locally compact group of polynomial growth having no non-trivial compact normal subgroups can be embedded as a co-compact subgroup into a semidirect product of a connected, simply connected, nilpotent Lie group and a compact group. There is also a uniqueness statement for this extension.
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Recently we have shown a structure theorem for locally compact groups of polynomial growth. We give now some applications on various growth functions and relations to FC-G - series. In addition, we show some results on related classes of groups.
We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
The celebrated theorem of Gromov asserts that any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner and others. In this note, we give yet another proof of Gromov's theorem, along the lines of Shalom and Chifan--Sinclair, which is based on the analysis of reduced cohomology and Shalom's property H_FD.
We show that finitely generated groups with a polynomial dimension growth have Yu's property A and give an example of such groups.
The aim of this work is to show that on a locally compact, second countable, compactly generated group $G$ with polynomial growth and homogeneous dimension $d_h$, there exist a continuous, proper, negative definite function $\ell$ with polynomial growth dimension $d_\ell$ arbitrary close to $d_h$.
We characterize the virtually nilpotent finitely generated groups (or, equivalently by Gromov's theorem, groups of polynomial growth) for which the Domino Problem is decidable: These are the virtually free groups, i.e. finite groups, and those having $\Z$ as a subgroup of finite index.
A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.
We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support. This has implications to the structure of the space of polynomially growing harmonic functions.
This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group $G$ has an element whose normal closure is abelian and of finite index, then $G$ has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).
In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups $U_ζ$ (at roots of unity $ζ$). These bounds depend only on the root system, and not on the characteristic $p$ or the size of the root of unity $ζ$. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system $Φ$. For …
Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric condition that guarantees that it agrees with the usual group cohomology and verify this condition for a class of combable groups. Our condition involves a chain complex that is closely related to exotic cohomology theories studied by Allcock and Gersten and by Mineyev.
Strongly Primitive Salem Growth Polynomials for Right-Angled Coxeter Groups
通过定义图的团多项式研究直角Coxeter群的标准球面增长率，证明每个偶数度至少4可作为强本原Salem增长率的度数，并给出独立多项式对指定Salem多项式的阻碍。
This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}(F_{\tt n})$, the outer automorphism group of a nonabelian free group $F_{\tt n}$. We prove that, for every subgroup $H$ of $\mathrm{Out}(F_{\tt n})$, there exists an element $ϕ\in H$ such that, for every element $g$ of $F_{\tt n}$, the conjugacy class $[g]$ has polynomial growth under iteration of $ϕ$ if and only if $[g]$ has polynomial growth under iteration of every element of $H$.
We prove that a finitely generated soluble residually finite group has polynomial index growth if and only if it is a minimax group. We also show that if a finitely generated group with PIG is residually finite-soluble then it is a linear group. These results apply in particular to boundedly generated groups; they imply that every infinite BG residually finite group has an infinite linear quotient.
Let G be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if G has polynomial growth, then (U^n) is a Folner sequence: that is, the volume of the boundary of U^n divided by U^n goes to zero. Moreover, we give a polynomial estimate of this ratio. Our proof is based on doubling property. As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a balls averages L^p-pointwise ergodic theorem for probability G-spaces, with G of polynomial growth and f…
We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy $\{f(n)\}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that $Cn^{d-2}$ firefighters are enough, answering and strengthening a conjecture of Develin and Hartke. We also prove that intermediate growth Cayley graphs do not satisfy polynomial containment, and give explicit lower bounds depending on the growth rate of the group. These bounds can be further improved when more geometric information is available, such as for Grigo…
Schramm's Locality Conjecture asserts that the value of the critical percolation parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this note, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon.
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 nu…
Asymptotic shape for subadditve processes on groups of polynomial growth
本研究探讨了虚拟幂零群上次可加过程的极限形状定理，在代数结构框架内提出了该定理的广义形式，扩展了次可加遍历理论以涵盖至多和至少线性随机增长过程，并给出了应用与示例。
Torsion homology growth of polynomially growing free-by-cyclic groups
本研究证明多项式增长单值群的自由循环群同调挠增长各维度均为零且与Farber链选取无关，推导出整挠等于ℓ²-挠，验证了Lück的相关猜想。
Homology growth of polynomially growing mapping tori
本文研究多项式增长自同构的映射圆柱的同调增长性质，证明其具有廉价重建性质并显示torsion同调增长在Farber序列下消失。
We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized $\ov{FC}$-groups. The following conditions turn out to be equivalent when $G$ is a metrizable compactly generated totally disconnected locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$; (iii) every inner automorphism of $G$ is distal; (iv) the contracti…
We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.
We consider a very simple Mealy machine (three states over a two-symbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function of its balls behaves asymptotically like n^2.4401..., where this constant is 1 + log(2)/log((1+sqrt(5))/2); that the semigroup satisfies the identity g^6=g^4; and that its lattice of two-sided ideals is a chain.
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