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 G , a word w w in the free group on k k generators induces a word map from G k G^{k} to G G . We say that w w is measure preserving with respect to G G if given uniform distribution on G k G^{k} , the image of this word map distributes uniformly on G 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 \mathbf {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 \mathbf {F}_{k} form a closed set in this topology.

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 Γ be a random d-regular graph on n vertices, and let λ be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [Alo86] that a random d-regular graph is "almost Ramanujan", in the following sense: for every ε > 0, λ < 2 √ d − 1 + ε asymptotically almost surely. Friedman famously presented a proof of this conjecture in [Fri08]. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies λ < 2 √ 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 Ω, we study the spectrum of Γ, a random n-covering of Ω. Let λ be the largest absolute value of a non-trivial eigenvalue of Γ. Extending Alon's conjecture to this more general model, Friedman [Fri03] conjectured that for every ε > 0, a.a.s. λ < ρ + ε, where ρ is the spectral radius of the universal cover of Ω. When Ω is regular we get a bound of ρ + 0.84, and for an arbitrary Ω, we prove a nearly optimal upper bound of √ 3ρ. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

ABSTRACT:We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ 1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19-35) Friedman (2008). has famously proved the (nearly?) optimal bound of 2Central to our work is a new analysis of formal words. Let w be a formal word in letters gThe word map associated with w maps the permutations σ 1 , . . . , σ k ∈ S n to the permutation obtained by replacing for each i, every occurrence of g i in w by σ i . We investigate the random variable X (n) w that counts the fixed points in this permutation when the σ i are selected uniformly at random. The analysis of the expectation E(X (n) w ) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture.As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703-730), which determines, for every fixed w, the limit distribution (as n → ∞) of X (n) w . 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.

Let F k be the free group on k generators. A word w ∈ 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 ≤ J ≤ 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 ∈ F k and consider the word map w : G × . . . × G → 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 × . . . × 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 .

Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ = 2 √ k − 1. We show that for every r, there is an r-covering (a.k.a. an r-lift) of G where all the new eigenvalues are bounded from above by ρ. It follows that a bipartite Ramanujan graph has a Ramanujan r-covering for every r. This generalizes the r = 2 case due to Marcus, Spielman and Srivastava [MSS15a].Every r-covering of G corresponds to a labeling of the edges of G by elements of the symmetric group S r . We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist.In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by [MSS15a], a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from [MSS15b].Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the r-th matching polynomial of G to be the average matching polynomial of all r-coverings of G. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [−ρ, ρ].