Uniform Exponential Growth for Linear Groups

Groups with near exponential residual finiteness growth

A function $\mathbb{N} \to \mathbb{N}$ is near exponential if it is bounded above and below by functions of the form $2^{n^c}$ for some $c > 0$. In this article we develop tools to recognize the near exponential residual finiteness growth in groups acting on rooted trees. In particular, we show the near exponential residual finiteness growth for certain branch groups, including the first Grigorchuk group, the family of Gupta-Sidki groups and their variations, and Fabrykowski-Gupta groups. We also show that the family of Gupta-Sidki p-groups, for $p\geq 5$, have super-exponential residual finit…

Solvable Groups of Exponential Growth and HNN Extensions

We prove the following version of Milnor's theorem on solvable groups of exponential growth: A finitely generated solvable group which is not polycyclic contains an ascending HNN extension. Consequently, a finitely generated solvable group is ERF iff it is polycyclic.

Uniform Growth, Actions on Trees and $GL_2$

We use actions on trees to determine uniform exponential growth for subgroups of $GL_2$.

一个非一致指数增长的圆柱双曲群

An acylindrically hyperbolic group of non-uniform exponential growth

构造了一个具有非一致指数增长和Kazhdan性质(T)的圆柱双曲群，该群是递增有限秩自由群的自同构群的公共商。

Exponential growth rates of free and amalgamated products

We prove that there is a gap between $\sqrt{2}$ and $(1+\sqrt{5})/2$ for the exponential growth rate of free products $G=A*B$ not isomorphic to the infinite dihedral group. For amalgamated products $G=A*_C B$ with $([A:C]-1)([B:C]-1)\geq2$, we show that lower exponential growth rate than $\sqrt{2}$ can be achieved by proving that the exponential growth rate of the amalgamated product $\mathrm{PGL}(2,\mathbb{Z})\cong (C_2\times C_2) *_{C_2} D_6$ is equal to the unique positive root of the polynomial $z^3-z-1$. This answers two questions by Avinoam Mann [The growth of free products, Journal of A…

Non proper HNN extensions and uniform uniform exponential growth

If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.

The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

We introduce a new invariant of finitely generated groups, the ambiguity function, and prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate $\lim \limits_{n \rightarrow \infty} \sqrt[n]{\vert B^{X}_H(n) \vert}$ of a subgroup $H$ of an acylindrically hyperbolic group $G$ exists with respect to every finite generating set $X$ of $G$, if $H$ contains a loxodromic element of $G$. Further we prove that the relative exponential growth rate of every finitely generated subgro…

Milnor's Problem on the Growth of Groups and its Consequences

We present a survey of results related to the Milnor's problem on group growth. We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but the main part of the article is devoted to the intermediate (between polynomial and exponential) growth case. A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) is considered, and a number of open problems is suggested.

On the conjugacy growth functions of groups

To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in "normal" cases, groups with exponential growth functions also have exponential conjugacy growth functions.

超限增长在小取消群中

Uniform growth in small cancellation groups

研究探讨了超限指数增长在超限小取消群中的闭包性质，证明了在超限小取消商下，此类群的闭包，并给出了超限指数增长与超限熵-基数估计的相关结论。

双曲空间上具有正常乘积作用的群的均匀指数增长

Uniform exponential growth for groups with proper product actions on hyperbolic spaces

本文研究有限生成群在双曲空间有限乘积上的正常作用，证明在粗稠密轨道或阴影性质下非虚阿贝尔子群具有均匀指数增长，并在弱非柱状条件下给出乘积集增长的完整分类。

Minimal exponential growth rates of metabelian Baumslag-Solitar groups and lamplighter groups

We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidire…

A finitely generated branch group of exponential growth without free subgroups

We will give an example of a branch group $G$ that has exponential growth but does not contain any non-abelian free subgroups. This answers question 16 from \cite{Bartholdi} positively. The proof demonstrates how to construct a non-trivial word $w_{a,b}(x,y)$ for any $a,b \in G$ such that $w_{a,b}(a,b) = 1$. The group $G$ is not just-infinite. We prove that every normal subgroup of $G$ is finitely generated as an abstract group and every proper quotient soluble. Further, $G$ has infinite virtual first Betti number but is not large.

On level-transitivity and exponential growth

We prove that if the group generated by a Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth for a restricted family of automaton semigroups.

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for $δ$-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson-Courtois-Gallot.

Uniformly exponential growth and mapping class groups of surfaces

We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.

Hierarchically hyperbolic groups and uniform exponential growth

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterization of hierarchically hyperbolic groups without uniform exponential growth. To achieve this, we gain new insights on the structure of certain classes of hierarchically hyperbolic groups. Our methods give a new unified proof of uniform exponential growth for several examples of gro…

A Wilson Group of Non-Uniformly Exponential Growth

This note constructs a finitely generated group $W$ whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 -- in other words, of non-uniformly exponential growth. This answers a question by Mikhael Gromov. The construction also yields a group of intermediate growth $V$ that locally resembles $W$ in that (by changing the generating set of $W$) there are isomorphic balls of arbitrarily large radius in $V$ and $W$'s Cayley graphs.

Extensions of hyperbolic groups have locally uniform exponential growth

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that…

Exponential growth of some iterated monodromy groups

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with…

作用于具有一致指数增长的 CAT(0) 立方复形的群

Groups acting on CAT(0) cube complexes with uniform exponential growth

研究作用于 CAT(0) 立方复形的群的一致指数增长，证明无全局不动点作用于方复形的群有一致指数增长或稳定欧几里得子复形，推广前人工作并应用于 Higman 群和无三角形 Artin 群子群。

Hyperconvex representations and exponential growth

Let $G$ be a real algebraic semi-simple Lie group and $Γ$ be the fundamental group of a compact negatively curved manifold. In this article we study the limit cone, introduced by Benoist, and the growth indicator function, introduced by Quint, for a class of representations $ρ:Γ\to G$ admitting a equivariant map from $\partialΓ$ to the Furstenberg boundary of $G$'s symmetric space together with a transversality condition. We then study how these objects vary with the representation.

1-余弦的亚线性增长与均匀凸性

Sublinear growth of 1-cocycles and uniform convexity

研究1-余弦在有限生成群上的亚线性增长特性，证明其与Banach空间的均匀凸性相关联。

A gap property for the growth of closed 3-manifold groups

We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented.

关于具有指数图增长的传递置换群

On transitive permutation groups with exponential graph growth

本文研究传递置换群的图增长问题，证明当群存在非平凡块且其补集的逐点稳定子非平凡时，该群的图增长必为指数级，从而推广了多个已有结果。

Purely exponential growth of cusp-uniform actions

Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal'bo-Otal-Peigné. For geometrically finite Cartan-Hadamard manifolds with pinched negative curvature this condition ensures the finiteness of Bowen-Margulis-Sullivan measures. In this case, our result recovers a theorem of Roblin (in a weaker form). Our main tool is the Patterson-Sullivan measures on the Gromov boundary of $X$…