arXiv subjects
Publications and source records attributed to Doron Puder.
We prove an analog of Aldous' spectral gap conjecture in the generalized symmetric groups $G\wr S_n$ where $G$ is an arbitrary finite group. Moreover, we show that Caputo's extension of the conjecture to hypergraphs transfers to these groups whenever it holds in the ordinary symmetric group.
Aldous' spectral gap conjecture, proven by Caputo, Liggett and Richthammer, states the following: for any set of transpositions in the symmetric group $\mathrm{Sym}(n)$, the spectral gap of the corresponding random walk on the group -- an $n!$-state process -- coincides with that of the corresponding random walk of a single element -- an $n$-state process. This paper presents an analog of this conjecture in the unitary group $\mathrm{U}(n)$, and proves it in several non-trivial cases. The phenomenon we discover is that for some natural families of probability distributions on $\mathrm{U}(n)$, the spectral gap of the corresponding random walk, which has a continuous state space, is identical to that of a discrete KMP process (also known as the uniform reshuffling process) with two indistinguishable particles on a hypergraph on $n$ vertices -- a discrete Markov chain with $\binom{n+1}{2}$ states.
Let $w$ be a word in a free group. As was revealed by Magee and Puder in [arXiv:1802.04862], the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain stable Fourier coefficients of $w$-random unitary matrices. In the first part of the current work [arXiv:2311.17733], we demonstrated how this phenomenon is much broader: we proved more instances of such results and conjectured others. These new results and conjectures involved other topological invariants (relatives of scl) and different families of groups. In the current paper we further extend and support this theory. We provide another instance of the theory and prove that the stable primitivity rank, too, can be expressed in terms of stable Fourier coefficients of $w$-random elements of groups. We introduce concrete formulas for stable Fourier coefficients of $w$-random elements in the symmetric group $S_N$ and its generalizations in the form of the wreath products $G\wr S_N$ where $G$ is an arbitrary compact group. We also define new stable invariants related to these groups, and prove they give bounds to many of the stable Fourier coefficients. As an aside, we generalize to tuples of words a result of Puder and Parzanchevski [arXiv:1202.3269] about the expected number of fixed points of $w$-random permutations.
Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of $S_n$. We prove the strong convergence in probability as $n\to\infty$ of this sequence of random representations to the regular representation of $\Gamma$. As a consequence, for any closed hyperbolic surface $X$, with probability tending to one as $n\to\infty$, a uniformly random degree-$n$ covering space of $X$ has near optimal relative spectral gap -- ignoring the eigenvalues that arise from the base surface $X$. To do so, we show that the polynomial method of proving strong convergence can be extended beyond rational settings. To meet the requirements of this extension we prove two new kinds of results. First, we show there are effective polynomial approximations of expected values of traces of elements of $\Gamma$ under random homomorphisms to $S_n$. Secondly, we estimate the growth rates of probabilities that a finitely supported random walk on $\Gamma$ is a proper power after a given number of steps.
Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces.
Fix a finite field $K$ of order $q$ and a word $w$ in a free group $F$ on $r$ generators. A $w$-random element in $GL_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\ldots,g_r\in GL_N(K)$ and evaluating $w\left(g_1,\ldots,g_r\right)$. Consider $\mathbb{E}_w\left[\mathrm{fix}\right]$, the average number of vectors in $K^{N}$ fixed by a $w$-random element. We show that $\mathbb{E}_{w}\left[\mathrm{fix}\right]$ is a rational function in $q^{N}$. Moreover, if $w=u^{d}$ with $u$ a non-power, then the limit $\lim_{N\to\infty}\mathbb{E}_{w}\left[\mathrm{fix}\right]$ depends only on $d$ and not on $u$. These two phenomena generalize to all stable characters of the groups $\left\{ GL_N(K)\right\}_{N}$. A main feature of this work is the connection we establish between word measures on $GL_N(K)$ and the free group algebra $K\left[F\right]$. A classical result of Cohn [1964] and Lewin [1969] is that every one-sided ideal of $K\left[F\right]$ is a free $K\left[F\right]$-module with a well-defined rank. We show that for $w$ a non-power, $\mathbb{E}_{w}\left[\mathrm{fix}\right]=2+\frac{C}{q^{N}}+O\left(\frac{1}{q^{2N}}\right)$, where $C$ is the number of rank-2 right ideals $I\le K\left[F\right]$ which contain $w-1$ but not as a basis element. We describe a full conjectural picture generalizing this result, featuring a new invariant we call the $q$-primitivity rank of $w$. In the process, we prove several new results about free group algebras. For example, we show that if $T$ is any finite subtree of the Cayley graph of $F$, and $I\le K\left[F\right]$ is a right ideal with a generating set supported on $T$, then $I$ admits a basis supported on $T$. We also prove an analogue of Kaplansky's unit conjecture for certain $K\left[F\right]$-modules.
For a fixed prime power $q$, let $\text{GL}_\bullet(q)$ denote the family of groups $\text{GL}_N(q)$ for $N \in \mathbb{Z}_{\geq 0}$. In this paper we study the $\mathbb{C}$-algebra of "stable" class functions of $\text{GL}_\bullet(q)$, and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple $\mathrm{VI}$-modules in the sense of [arXiv:1408.3694,arXiv:1602.00654]. A second one comes from characters of parabolic representations. The final two, one originally defined in [arXiv:1803.04155] and the other in [arXiv:2110.11099], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [arXiv:2106.11587].
What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non-backtracking random walk. This generalizes the well-known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree.
Let $w$ be a word in a free group. A few years ago, Magee and the first named author discovered that the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain Fourier coefficients of $w$-random unitary matrices [arXiv:1802.04862]. But the random-matrix side of this equality can be naturally tweaked by considering $w$-random permutations, $w$-random orthogonal matrices and so on, to produce new invariants for any given word. Are these invariants new? interesting? Do they admit an intrinsic topological description as in the case of $w$-random unitaries and scl? The current paper formalizes the definition of these invariants coming from $w$-random matrices, answers the above questions in certain cases involving generalized symmetric groups, and poses detailed conjectures in many others. In particular, we present a plethora of topological, combinatorial and algebraic invariants of words which play, or are at least conjectured to play, a similar role to the one played by scl in the above-mentioned result. Among others, these invariants include two invariants recently defined by Wilton [arXiv:2210.09853]: the stable primitivity rank and a non-oriented analog of scl.
In their celebrated paper (arXiv:0906.1238), Caputo, Liggett and Richthammer proved Aldous' conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality - a certain inequality of operators in the group ring $\mathbb{R}[\mathrm{Sym}_n]$ of the symmetric group. Here we generalize the Octopus Inequality and apply it to generalize the Caputo-Liggett-Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.
Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ has a convergent Laurent expansion at $n=\infty$ involving maps on surfaces and $L^{2}$-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning $\mathrm{U}\left(n\right)$, yet with some surprising twists. A priori to our result, ${\cal T}r_{w}^{\mathrm{O}}(n)$ does not change if $w$ is replaced with $α(w)$ where $α$ is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under $\mathrm{Aut}(\mathrm{\mathbf{F}}_{r})$. As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of ${\cal T}r_{w}^{\mathrm{O}}(n)$ as $n\to\infty$, we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani. Our results are obtained more generally for a tuple of words $w_1,\ldots,w_\ell$, leading to functions ${\cal T}r_{w_{1},\ldots,w_{\ell}}^{\mathrm{O}}$. We also obtain all the analogous results for the compact symplectic groups $\mathrm{Sp}\left(n\right)$ through a rather mysterious duality formula.
Let $α$ and $β$ be uniformly random permutations of orders $2$ and $3$, respectively, in $S_{N}$, and consider, say, the permutation $αβαβ^{-1}$. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let $Γ=G_{1}*\ldots*G_{k}$ be a free product of groups where each of $G_1,\ldots,G_k$ is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element $γ\inΓ$, a $γ$-random permutation in the symmetric group $S_{N}$ is the image of $γ$ through a uniformly random homomorphism $Γ\to S_{N}$. In this paper we study local statistics of $γ$-random permutations and their asymptotics as $N$ grows. We first consider $\mathbb{E}\left[\mathrm{fix}_γ\left(N\right)\right]$, the expected number of fixed points in a $γ$-random permutation in $S_{N}$. We show that unless $γ$ has finite order, the limit of $\mathbb{E}\left[\mathrm{fix}_γ\left(N\right)\right]$ as $N\to\infty$ is an integer, and is equal to the number of subgroups $H\leΓ$ containing $γ$ such that $H\cong\mathbb{Z}$ or $H\cong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $H\leΓ$ containing $γ$ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $\mathbb{E}\left[\mathrm{fix}_γ\left(N\right)\right]$ and determine the limit distribution of the number of fixed points as $N\to\infty$. These results are then generalized to all statistics of cycles of fixed lengths.
Let $Γ_g$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We introduce a combinatorial structure of "core surfaces", that represent subgroups of $Γ_g$. These structures are (usually) 2-dimensional complexes, made up of vertices, labeled oriented edges, and $4g$-gons. They are compact whenever the corresponding subgroup is finitely generated. The theory of core surfaces that we initiate here is analogous to the influential and fruitful theory of Stallings core graphs for subgroups of free groups.
Let $X$ be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each $n\in\mathbf{N}$, let $X_{n}$ be a random degree-$n$ cover of $X$ sampled uniformly from all degree-$n$ Riemannian covering spaces of $X$. An eigenvalue of $X$ or $X_{n}$ is an eigenvalue of the associated Laplacian operator $Δ_{X}$ or $Δ_{X_{n}}$. We say that an eigenvalue of $X_n$ is new if it occurs with greater multiplicity than in $X$. We prove that for any $\varepsilon>0$, with probability tending to 1 as $n\to\infty$, there are no new eigenvalues of $X_n$ below $\frac{3}{16}-\varepsilon$. We conjecture that the same result holds with $\frac{3}{16}$ replaced by $\frac{1}{4}$.
Let $Γ_{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We develop a new method for integrating over the representation space $\mathbb{X}_{g,n}=\mathrm{Hom}(Γ_{g},S_{n})$ where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots,n\}$. Equivalently, this is the space of all vertex-labeled, $n$-sheeted covering spaces of the the closed surface of genus $g$. Given $ϕ\in\mathbb{X}_{g,n}$ and $γ\inΓ_{g}$, we let $\mathsf{fix}_γ(ϕ)$ be the number of fixed points of the permutation $ϕ(γ)$. The function $\mathsf{fix}_γ$ is a special case of a natural family of functions on $\mathbb{X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to\infty$, for the expectation of $\mathsf{fix}_γ$ with respect to the uniform probability measure on $\mathbb{X}_{g,n}$, which is denoted by $\mathbb{E}_{g,n}[\mathsf{fix}_γ]$. We prove that if $γ\inΓ_{g}$ is not the identity, and $q$ is maximal such that $γ$ is a $q$th power in $Γ_{g}$, then \[ \mathbb{E}_{g,n}[\mathsf{fix}_γ]=d(q)+O(n^{-1}) \] as $n\to\infty$, where $d\left(q\right)$ is the number of divisors of $q$. Even the weaker corollary that $\mathbb{E}_{g,n}[\mathsf{fix}_γ]=o(n)$ as $n\to\infty$ is a new result of this paper. We also prove that if $γ$ is not the identity then $\mathbb{E}_{g,n}[\mathsf{fix}_γ]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$.
The main goal of this note is to illustrate the advantage of analyzing the non-backtracking spectrum of a regular graph rather than the ordinary spectrum. We show that by switching to non-backtracking spectrum, the method of proof used in [Puder 2015, arXiv::1212.5216] yields a bound of $2\sqrt{d-1}+\frac{2}{\sqrt{d-1}}$ instead of the original $2\sqrt{d-1}+1$ on the second largest eigenvalue of a random $d$-regular graph.
Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating $w\left(\sigma_{1},\ldots,\sigma_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a $w$-random permutation is $1+\theta\left(N^{1-\pi\left(w\right)}\right)$, where $\pi\left(w\right)$ is the smallest rank of a subgroup $H\le F$ containing $w$ as a non-primitive element. We show that $\pi\left(w\right)$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge2$, the average number of $t$-cycles is $\frac{1}{t}+O\left(N^{-\pi\left(w\right)}\right)$. As an application, we prove that for every $s$, every $\varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have second eigenvalue at most $2\sqrt{2r-1}+\varepsilon$ asymptotically almost surely. An important ingredient in this work is a systematic study of not-necessarily connected Stallings core graphs.
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 $d\in\mathbb{Z}$.