首页/论文/1902.04873

arXiv 1902.048732019-02math.GRmath.CO

Surface Words are Determined by Word Measures on Groups

Michael Magee, Doron Puder

备注：: 16 pages, fixed the proof of Theorem 3.6, updated references
发表信息：: Israel Journal of Mathematics 241, 749--774, 2021
DOI：: 10.1007/s11856-021-2113-5

英文摘要：

Every word $w$ in a free group naturally induces a probability measure on every compact group $G$. For example, if $w=\left[x,y\right]$ is the commutator word, a random element sampled by the $w$-measure is given by the commutator $\left[g,h\right]$ of two independent, Haar-random elements of $G$. Back in 1896, Frobenius showed that if $G$ is a finite group and $ψ$ an irreducible character, then the expected value of $ψ\left(\left[g,h\right]\right)$ is $\frac{1}{ψ\left(e\right)}$. This is true for any compact group, and completely determines the $\left[x,y\right]$-measure on these groups. An analogous result holds with the commutator word replaced by any surface word. We prove a converse to this theorem: if $w$ induces the same measure as $\left[x,y\right]$ on every compact group, then, up to an automorphism of the free group, $w$ is equal to $\left[x,y\right]$. The same holds when $\left[x,y\right]$ is replaced by any surface word. The proof relies on the analysis of word measures on unitary groups and on orthogonal groups, which appears in separate papers, and on new analysis of word measures on generalized symmetric groups that we develop here.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Some Orbits of Free Words that are Determined by Measures on Finite Groups

Every word in a free group $F$ induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of $\mathrm{Aut}F$. A special case of this problem, when one of the words is the primitive word $x$, was settled positively by the third author and Parzanchevski [arXiv:1202.3269]. Here we extend this result to the case where one of the words is $x^d$ or $\left[x,y\right]^{d}$ for an arbitrary $d\in\mathbb{Z}$.

2020-07-29math.GR

On the profinite rigidity of surface groups and surface words

Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.

2020-10-15math.GRmath.GT

The word problem for $κ$-terms over the pseudovariety of local groups

In this paper we study the $κ$-word problem for the pseudovariety ${\bf LG}$ of local groups, where $κ$ is the canonical signature consisting of the multiplication and the pseudoinversion. We solve this problem by transforming each arbitrary $κ$-term $α$ into another one called the canonical form of $α$ and by showing that different canonical forms have different interpretations over ${\bf LG}$. The procedure of construction of these canonical forms consists in applying elementary changes determined by a certain set $Σ$ of $κ$-identities. As a consequence, $Σ$ is a basis of $κ$-identities for …

2015-09-04math.GR

Local similarity groups with context-free co-word problem

Let $G$ be a group, and let $S$ be a finite subset of $G$ that generates $G$ as a monoid. The co-word problem is the collection of words in the free monoid $S^{\ast}$ that represent non-trivial elements of $G$. A current conjecture, based originally on a conjecture of Lehnert and modified into its current form by Bleak, Matucci, and Neuhöffer, says that Thompson's group $V$ is a universal group with context-free co-word problem. In other words, it is conjectured that a group has a context-free co-word problem exactly if it is a finitely generated subgroup of $V$. Hughes introduced the class $\…

2014-06-18math.GR

Measure Preserving Words are Primitive

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces a word map from $G^k$ to $G$. We say that $w$ is measure preserving with respect to $G$ if given uniform distribution on $G^k$, the image of this word map distributes uniformly on $G$. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectur…

2014-10-23math.COmath.GRmath.PR

On certain C-test words for free groups

Let F_m be a free group of a finite rank m > 1 and X_i, Y_j be elements in F_m. A non-empty word w(x_1,..., x_n) is called a C-test word in n letters for F_m if, whenever w(X_1,..., X_n)=w(Y_1,..., Y_n) not equal to 1, the two n-tuples (X_1,..., X_n) and (Y_1,..., Y_n) are conjugate in F_m. In this paper we construct, for each n > 1, a C-test word v_n(x_1,..., x_n) with the additional property that v_n(X_1,..., X_n)=1 if and only if the subgroup of F_m generated by X_1,..., X_n is cyclic. Making use of such words v_m(x_1,..., x_m) and v_{m+1}(x_1,..., x_{m+1}), we provide a positive solution t…

2001-03-16math.GR

更多相关论文收起更多论文

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil.…

2011-12-22cs.DMmath.COmath.GR

Word Measures on Unitary Groups

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free group $F_r$ on $r$ generators determines a word map from $U(n)^r$ to $U(n)$, defined by substitutions. The $w$-measure on $U(n)$ is defined as the pushforward via this word map of the Haar measure on $U(n)^r$. Let $Tr_w(n)$ denote the expected trace of a random unitary matrix sa…

2016-02-01math.GRmath.GT

On the Distribution of Values of Certain Word Maps

For any positive integers m and n, the word map (x,y) -> x^m y^n is almost measure preserving on large finite simple groups G.

2013-08-06math.GR

圈积上的字测度 I

Word Measures On Wreath Products I

本文通过逼近圈积 $G\wr S_n$ 不可约特征标的 $w$-期望，揭示了推广本原秩 $π(w)$ 的字自同构不变量，并证明了其多项式衰减性质及在树自同构群中的应用。

2023-05-18math.GR

词测度在圈积上的研究 II

Word Measures on Wreath Products II

本文针对圈积 $G\wr S_n$ 上的稳定特征标，以 $O(n^{-\pi(w)})$ 误差逼近词 $w$ 的期望，推广先前结果，并证明表示稳定作用的随机 Schreier 图为近最优扩展图，揭示稳定表示理论与 Stallings 核图的联系。

2026-09-22math.COmath.GRmath.RT

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…

2008-07-15math.DSmath.GR

Word-Induced Measures on Compact Groups

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure μ_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that μ_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsen's classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for μ_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These …

2011-02-21math.GNmath.GRmath.PR