On the continuity of the growth rate on the space of Coxeter systems

Subgroup growth of right-angled Artin and Coxeter groups

We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.

Salem 数、谱半径与双曲 Coxeter 群的增长率

Salem numbers, spectral radii and growth rates of hyperbolic Coxeter groups

本文研究了双曲 Coxeter 群的增长率与 Salem 数、谱半径之间的关系，证明了并非所有 Salem 数都对应于紧致双曲 Coxeter 群的增长率，并探讨了平面与三面体 Coxeter 群的增长率特性。

Growth rates of Coxeter groups and Perron numbers

We define a large class of abstract Coxeter groups, that we call $\infty$--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We also show that for this class the geodesic growth rate strictly dominates the word growth rate.

具有2维Davis复形的Coxeter系统、增长率与Perron数

Coxeter systems with $2$-dimensional Davis complexes, growth rates and Perron numbers

本文研究低维Coxeter系统的增长率，证明其神经欧拉示性数的符号对应增长率为Salem、Pisot数，并构造了负欧拉示性数下增长率为Perron数的无穷多非双曲Coxeter系统，推广了已有结果。

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal $1$. Also, we compute the average number of geodesics representing an element of given word length in such groups.

关于Coxeter群的生成函数

On Growth Functions of Coxeter Groups

本文研究了秩$n$的Coxeter系统的生成函数有限性问题，证明了$2$-球面条件下$q=n-1$时有限性成立，并在Coxeter图是完全图时$q=n-2$时无限，从而完整刻画了该情形下的有限性。

直角Coxeter群的强本原Salem增长多项式

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。

Coxeter群、图积及其他群中的压缩元素与共轭增长

Contracting elements and conjugacy growth in Coxeter groups, graph products, and further groups

本文在periagroups的标准Cayley图中构造压缩元素，证明除非本质分裂为直积，否则这些群是圆柱双曲的且共轭增长级数超越。

标记Dyer系统空间、增长速率的单调性和连续性

The space of marked Dyer systems, monotonicity, and continuity of growth rates

本文研究了n-标记Dyer系统空间，证明其增长速率在特定偏序下单调递增并连续，扩展了Coxeter系统的连续性结果。

Geodesic growth of right-angled Coxeter groups based on trees

In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $T^1_n$ and $T^2_n$ have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree does is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $\{S^1_n\}_{n \geq 11}$ and $\{S^2_n\}_{n \geq 11}$, on $n$ vertices, such that $S^1_n$ and…

Data about hyperbolic Coxeter systems

We collect several data about Coxeter systems (cf. [Bou07, Hum90]), with particular emphasis on the hyperbolic ones. For each ($\preceq$-minimal) hyperbolic Coxeter system (W,S) the Poincaré series \[p_{(W,S)}(t)=\sum_{w\in W} t^{\ell(w)}\] and the growth rate \[ ω(W,S)=\limsup_n \sqrt[n]{a_n}\] are explicitly computed using Magma (cf. [BCP97]). These computations were performed in connection to the proof of [Ter, Thm. B]. Since the Poincaré series represents a rational function, one may recover the sequence $(a_k)_{k\geq 0}$ through a linear recurrence relation on the coefficients, provided t…

Geodesic Growth of some 3-dimensional RACGs

We give explicit formulas for the geodesic growth series of a Right Angled Coxeter Group (RACG) based on a link-regular graph that is 4-clique free, i.e. without tetrahedrons

多项式增长映射圆柱的同调增长

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。

On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual re…