On bi-invariant word metrics

SL(2,C)^2中某些词变种的ML不变量和自同构群

ML-invariant and automorphism groups of certain word varieties in SL(2,C)^2

研究SL(2,C)^2中某些词变种的ML不变量和自同构群，证明其为Jordan群。

Heegaard Floer同调与Torelli群的词度量

Heegaard Floer homology and the word metric on the Torelli group

研究Heegaard Floer同调修正项与Torelli群词度量的关系，证明Torelli群Cayley图在特定生成集下的无限直径，同时展示Torelli群子集在该度量下的有界性，并排除有理同调球的Morita型公式。

Entropy and drift for word metric on relatively hyperbolic groups

We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blach{è}re, Ha{ï}ssinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hyperbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect t…

Invariant measures on the space of horofunctions of a word hyperbolic group

We introduce a natural equivalence relation on the space $\sH_0$ of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if $η$ is such a measure and $G$ acts on a space $(X,μ)$ by p.m.p. transformations then $η\times μ$ is virtually ergodic with respect to a natural equivalence relation on $\sH_0\times X$. This is comparable to a special case of the Howe-Moore theorem. These results are a…

相消范数与双不变字度量的几何

Cancelation norm and the geometry of biinvariant word metrics

本文研究群上的双不变字度量，提出了计算自由群双不变字范数的高效算法，构造了局部紧树到非交换自由群双不变Cayley图的等距嵌入，并证明了几何起源群类中循环子群的bq-二分法性质。

Metric characterizations of superreflexivity in terms of word hyperbolic groups and finite graphs

We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups. We compare characterizations of superreflexivity in terms of diamond graphs and binary trees. We show that there exist sequences of series-parallel graphs of increasing topological complexity which admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superreflexivity.

从随机矩阵中稳定的不变量

Stable Invariants of Words from Random Matrices

本文研究了通过随机矩阵定义的新的不变量，探讨其拓扑意义，并提出多项猜想，包括稳定原始秩和非定向scl的类比。

(ℤ/2ℤ)≀𝔽_d的字长谱三元组不是度量

Word-Length Spectral Triples of $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$ Are Not Metric

本文针对非交换度量几何中紧量子度量空间的判定问题，以灯夫群(ℤ/2ℤ)≀𝔽_d为研究对象，证明其典范谱三元组无法成为谱度量空间，给出该领域首个明确的反例族。

The word problem and the metric for the Thompson-Stein groups

We consider the Thompson-Stein group F(n_1,...,n_k) for integers n_1,...,n_k and k greater than 1. We highlight several differences between the cases k=1$ and k>1, including the fact that minimal tree-pair diagram representatives of elements may not be unique when k>1. We establish how to find minimal tree-pair diagram representatives of elements of F(n_1,...,n_k), and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a unique normal form for elements of F(n_1,...,n_k) (with respect to the standard infinite generating set developed by Melanie St…

Images of word maps in finite simple groups

In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset $A$ of a finite simple group $G$ is the image of some word map $w : G\times G\to G $ if and only if (i) $A$ contains the identity and (ii) $A$ is invariant under $\Aut (G)$.

Word images in symmetric and classical groups of Lie type are dense

Let $w\in\mathbf F_k$ be a non-trivial word and denote by $w(G)\subseteq G$ the image of the associated word map $w\colon G^k\to G$. Let $G$ be one of the finite groups ${\rm S}_n,{\rm GL}_n(q),{\rm Sp}_{2m}(q),{\rm GO}_{2m}^\pm(q),{\rm GO}_{2m+1}(q),{\rm GU}_n(q)$ ($q$ a prime power, $n\geq 2$, $m\geq 1$), or the unitary group ${\rm U}_n$ over $\mathbb C$. Let $d_G$ be the normalized Hamming distance resp. the normalized rank metric on $G$ when $G$ is a symmetric group resp. one of the other classical groups and write $n(G)$ for the permutation resp. Lie rank of $G$. For $\varepsilon>0$, we p…

Images of word maps in almost simple groups and quasisimple groups

It has been shown by Lubotzky in [10] that the set of verbal images of a fixed non-abelian finite simple group G is precisely the set of endomorphism invariant subsets of G. Here we use his result to determine the verbal images of certain almost simple groups and quasisimple groups.

不同秩的自由群是否双不变拟等距？

Are free groups of different ranks bi-invariantly quasi-isometric?

本文证明了装备双不变词度量的有限秩自由群间的同态为拟等距当且仅当其为同构。

度量意义下LEF的群

Groups which are metrically LEF

本文研究局部可嵌入性和LEF的度量版本，重点分析带字度量的正规有限生成群，推广了经典群论概念。

随机矩阵所得词的稳定不变量 II：公式与推广

Stable Invariants of Words from Random Matrices II: Formulas and Extensions

本文扩展随机矩阵与词的稳定不变量理论，证明稳定本原秩可由稳定 Fourier 系数表示，给出 $S_N$ 及圈积 $G\wr S_N$ 中的具体公式，并推广随机排列不动点期望结果。

关于具有EDT0L字问题的群

On groups with EDT0L word problem

该研究证明无限循环群的字问题不满足EDT0L条件，并指出有限生成的EDT0L字问题群必为torsion群，同时证明该性质在生成集和子群变化下保持不变。

Metric topological groups: their metric approximation and metric ultraproducts

We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hype…