arXiv subjects

Doron Puder

Publications and source records attributed to Doron Puder.

34 records · Page 2Linked to original sources

arXiv2020

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 $d\in\mathbb{Z}$.

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arXiv2020

Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n)

We consider closed meandric systems, and their equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions $NC(n)$. In this equivalent description, the number of components of a random meandric system of order $n$ translates into the distance between two partitions in $NC(n)$. We focus on a class of couples $(π,ρ)\in NC(n)^2$ -- namely the ones where $π$ is conditioned to be an interval partition -- for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a non-trivial class of meanders (i.e. connected meandric systems), which we call "meanders with shallow top", and which can be explicitly enumerated. Moreover, the expected number of components for a random "meandric system with shallow top", is asymptotically $(9n+28)/27$. Our calculations concerning expected number of components are related to the idea of taking the derivative at $t=1$ in a semigroup for the operation $\boxplus$ of free probability (but the underlying considerations are presented in a self-contained way, and can be followed without assuming a free probability background). Let $c_{n}'$ denote the expected number of components of a general, unconditioned, meandric system of order $n$. A variation of the methods used in the shallow-top case allows us to prove that $\mathrm{lim\ inf}_{n\to\infty}c_{n}'/n\geq0.17$. We also note that, by a direct elementary argument, one has $\mathrm{lim\ sup}_{n\to\infty}c_{n}'/n\leq0.5$. These bounds support the conjecture that $c_{n}'$ follows a regime of "constant times $n$" (where numerical experiments suggest that the constant should be $\approx0.23$).

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arXiv2020

Aldous' Spectral Gap Conjecture for Normal Sets

Let $S_n$ denote the symmetric group on $n$ elements, and $Σ\subseteq S_{n}$ a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $Σ$ is a set of transpositions, then the second eigenvalue of the Cayley graph $\mathrm{Cay}\left(S_{n},Σ\right)$ is identical to the second eigenvalue of the Schreier graph on $n$ vertices depicting the action of $S_{n}$ on $\left\{ 1,\ldots,n\right\}$. Inspired by this seminal result, we study similar questions for other types of sets in $S_{n}$. Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [2008], we show that for large enough $n$, if $Σ\subset S_{n}$ is a full conjugacy class, then the second eigenvalue of $\mathrm{Cay}\left(S_{n},Σ\right)$ is roughly identical to the second eigenvalue of the Schreier graph depicting the action of $S_{n}$ on ordered $4$-tuples of elements from $\left\{ 1,\ldots,n\right\}$. We further show that this type of result does not hold when $Σ$ is an arbitrary normal set, but a slightly weaker one does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set $Σ\subset S_{n}$, which yields surprisingly strong consequences.

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arXiv2019

Automorphism-invariant positive definite functions on free groups

In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new perspectives and interesting observations; therefore the style is expository.

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arXiv2019

Surface Words are Determined by Word Measures on Groups

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.

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arXiv2019

Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group $\mathcal{U}\left(n\right)$ of unitary matrices. More concretely, we study measures induced by free words on $\mathcal{U}\left(n\right)$. Let $F_{r}$ be the free group on $r$ generators. To sample a random element from $\mathcal{U}\left(n\right)$ according to the measure induced by $w\in F_{r}$, one substitutes the $r$ letters in $w$ by $r$ independent, Haar-random elements from $\mathcal{U}\left(n\right)$. The main theme of this paper is that every moment of this measure is determined by families of pairs $\left(Σ,f\right)$, where $Σ$ is an orientable surface with boundary, and $f$ is a map from $Σ$ to the bouquet of $r$ circles, which sends the boundary components of $Σ$ to powers of $w$. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of $Σ$. As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on $\mathcal{U}\left(n\right)$ bears information about the number of solutions to the equation $\left[u_{1},v_{1}\right]\cdots\left[u_{g},v_{g}\right]=w$ in the free group, and deduce that one can ``hear'' the stable commutator length of a word through its unitary word measures.

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arXiv2017

The Markoff Group of Transformations in Prime and Composite Moduli

The Markoff group of transformations is a group $Γ$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $Γ$ acts transitively on the set $X^{*}\left(p\right)$ of non-zero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of permutations obtained by the action of $Γ$ on $X^{*}\left(p\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $Γ$ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman, stating that for any finite non-abelian simple group $G$ and $r\ge3$, the group $\mathrm{Aut}\left(F_{r}\right)$ acts on at least one $T_{r}$-system of $G$ as the alternating or symmetric group. In this language, our main result translates to that for most primes $p$, the group $\mathrm{Aut}\left(F_{2}\right)$ acts on a particular $T_{2}$-system of $\mathrm{PSL}\left(2,p\right)$ as the alternating or symmetric group.

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arXiv2016

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 sampled from $U(n)$ according to the $w$-measure. It was shown by Voiculescu [Voic 91'] that for $w \ne 1$ this expected trace is $o(n)$ asymptotically in $n$. We relate the numbers $Tr_w(n)$ to the theory of commutator length of words and obtain a much stronger statement: $Tr_w(n)=O(n^{1-2g})$, where $g$ is the commutator length of $w$. Moreover, we analyze the number $\lim_{n\to\infty}n^{2g-1} \cdot Tr_w(n)$ and show it is an integer which, roughly, counts the number of (equivalence classes of) solutions to the equation $[u_1,v_1]...[u_g,v_g]=w$ with $u_i,v_i \in F_r$. Similar results are obtained for finite sets of words and their commutator length, and we deduce that one can 'hear' the stable commutator length of a word by 'listening' to its unitary measures.

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arXiv2015

Expansion of Random Graphs: New Proofs, New Results

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices, and let λbe the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [86'] that a random d-regular graph is almost Ramanujan, in the following sense: for every e>0, λ<2\sqrt{d-1} + e asymptotically almost surely. Friedman famously presented a proof of this conjecture in [08']. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies λ< 2\sqrt{d-1} + 1 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For d even, a d-regular graph on n vertices is an n-covering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph H, we study the spectrum of G, a random n-covering of H. Let λbe the largest absolute value of a non-trivial eigenvalue of G. Extending Alon's conjecture to this more general model, Friedman [03'] conjectured that for every e>0, a.a.s. λ< ρ+e, where ρis the spectral radius of the universal cover of H. When H is regular we get a bound of ρ+0.84, and for an arbitrary H, we prove a nearly optimal upper bound of \sqrt{3}ρ. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

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arXiv2014

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 conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups, and Möbius inversions. Our methods yield the stronger result that a subgroup of $F_k$ is measure preserving if and only if it is a free factor. As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of $F_k$ form a closed set in this topology.

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arXiv2014

Growth of Primitive Elements in Free Groups

In the free group $F_k$, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length $N$ contains one of the letters exactly once asymptotically almost surely (as $N \to \infty$). This also solves a question from the list `Open problems in combinatorial group theory' [Baumslag-Myasnikov-Shpilrain 02']. Let $p_{k,N}$ be the number of primitive words of length $N$ in $F_k$. We show that for $k \ge 3$, the exponential growth rate of $p_{k,N}$ is $2k-3$. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.

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arXiv2012

Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.

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arXiv2012

Stallings Graphs, Algebraic Extensions and Primitive Elements in F2

This paper studies the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

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arXiv2011

On Primitive Words I: A New Algorithm

Let $F_k$ be the free group on $k$ generators, and let $H\le J\le \F_k$ be subgroups of finite rank. We present a new elementary algorithm to determine whether $H$ is a free factor of $J$. In particular, this algorithm can determine whether a given element $w\in F_k$ is primitive, i.e. whether it belongs to some basis of $F_k$.

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arXiv2010

More on the phi = beta Conjecture and Eigenvalues of Random Graph Lifts

Let $G$ be a connected graph, and let $λ_1$ and $ρ$ denote the spectral radius of $G$ and the universal cover of $G$, respectively. In \cite{Fri03}, Friedman has shown that almost every $n$-lift of $G$ has all of its new eigenvalues bounded by $O(λ_1^{1/2}ρ^{1/2})$. In \cite{LP10}, Linial and Puder have improved this bound to $O(λ_1^{1/3}ρ^{2/3})$. Friedman had conjectured that this bound can actually be improved to $ρ+ o_n(1)$ (e.g., see \cite{Fri03,HLW06}). In \cite{LP10}, Linial and Puder have formulated two new categorizations of formal words, namely $ϕ$ and $β$, which assign a non-negative integer or infinity to each word. They have shown that for every word $w$, $ϕ(w) = 0$ iff $β(w) = 0$, and $ϕ(w) = 1$ iff $β(w) = 1$. They have conjectured that $ϕ(w) = β(w)$ for every word $w$, and have run extensive numerical simulations that strongly suggest that this conjecture is true. This conjecture, if proven true, gives us a very promising approach to proving a slightly weaker version of Friedman's conjecture, namely the bound $O(ρ)$ on the new eigenvalues (see \cite{LP10}). In this paper, we make further progress towards proving this important conjecture by showing that $ϕ(w) = 2$ iff $β(w) = 2$ for every word $w$.

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arXiv2009

Words Maps and Spectra of Random Graph Lifts

We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by s_i. We investigate the random variable X_w^n that counts the fixed points in this permutation when the s_i are selected uniformly at random. A major ingredient of our work is a new categorization of words which considerably extends the dichotomy of primitive vs. imprimitive words. We establish some results and make a few conjectures about the relation between the expectation E(X_w^n) and this new categorization. This analysis contributes deeply to our study of the spectra of random lifts of graphs. Let G be a connected graph, and let the infinite tree T be its universal cover space. If L and R are the spectral radii of G and T respectively, then, as shown by J. Friedman, for almost every n-lift H of G, all "new" eigenvalues of H are < O(L^(1/2)R^(1/2)). We improve this upper bound to O(L^(1/3)R^(2/3)), and our aforementioned conjectures suggest a possible approach to proving an upper bound of O(R). This is a generalization of the problem of bounding the second eigenvalue in a random 2d-regular graph. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica, which determines, for every fixed w, the limit distribution (as n \to \infty) of X_w^n. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w=u^d for some word u.

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