arXiv subjects

Itamar Vigdorovich

Publications and source records attributed to Itamar Vigdorovich.

14 recordsLinked to original sources

arXiv2026

Selflessness, MIF and opposition in groups acting on exotic buildings

We prove that groups acting freely and cocompactly on (possibly exotic) affine buildings of type $\tilde{A}_2$ and $\tilde C_2$ are mixed-identity free and have selfless reduced $C^*$-algebras. These results follow from a strong form of ping-pong dynamics that we call 'transversal contractivity'. Our main geometric result regards domesticity properties of elements in the associated polygons at infinity: we prove that, in our context, the opposite geometry of a hyperbolic element is topologically large.

math.GR↗

arXiv2026

On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

We show the following dichotomy for a connected Lie group $G$: If $G$ is amenable, then any topologically generating set $X\subset G$ of size larger than a fixed polynomial in the dimension of $G$ must be redundant (i.e., a proper subset of $X$ still generates $G$). If $G$ is non-amenable, then it admits arbitrarily large topologically generating sets that are irredundant, and remain irredundant even after applying Nielsen transformations. The polynomial bound for amenable groups is obtained by reduction to finite simple groups of Lie type via strong approximation. This partially answers two conjectures by Gelander on generation in compact Lie groups and simple algebraic groups, and moreover shows that these conjectures are implied by the Wiegold conjecture. The construction of large Nielsen irredundant generating sets in non-amenable groups is done by extending Minsky's work to higher rank Lie groups, exhibiting dense representations in the domain of discontinuity of the $\mathrm{Out}(F_{n})$-action on the character variety.

math.GR↗

arXiv2026

Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices

We prove that if the group $\mathrm{SL}_2(\mathbb Z[1/p])$ is flexibly Hilbert--Schmidt stable for some prime $p$, then it admits a non-hyperlinear finite central extension. Consequently, a positive answer to the following question would yield an explicit example of a non-hyperlinear group: If two representations of the modular group $\mathrm{SL}_2(\mathbb{Z})$ almost agree on an Iwahori subgroup $B$, must they be close to representations that agree on $B$? More generally, we investigate spectral gap properties for asymptotic representations of higher rank lattices and groups with property (T:FD). In this setting, we prove that character rigidity is equivalent to a weak form of stability.

math.GR↗

arXiv2026

Selfless reduced $C^{*}$-algebras of linear groups

It is shown that the reduced C*-algebra of a nontrivial linear group $Γ<GL_{d}(k)$ with trivial amenable radical is selfless. Thus selflessness and simplicity coincide for reduced C*-algebras of linear groups. Similar results are obtained for twisted reduced group C*-algebra.

math.OA↗

arXiv2026

Characters of surface groups

We initiate the study of characters of surface groups and their corresponding tracial representations. We show that any tracial representation can be approximated arbitrarily well in the Wasserstein topology by factorial tracial representations with spectral gap. In particular, we deduce that the space of traces of a surface group is the Poulsen simplex, thereby resolving positively a question posed by Orovitz, Slutsky, and the third author.

math.GR↗

arXiv2025

Lifting Generators in Connected Lie Groups

Given an epimorphism between topological groups $f:G\to H$, when can a generating set of $H$ be lifted to a generating set of $G$? We show that for connected Lie groups the problem is fundamentally abelian: generators can be lifted if and only if they can be lifted in the induced map between the abelianisations (assuming the number of generators is at least the minimal number of generators of $G$). As a consequence, we deduce that connected perfect Lie groups satisfy the Gaschütz lemma: generating sets of quotients can always be lifted. If the Lie group is not perfect, this may fail. The extent to which a group fails to satisfy the Gaschütz lemma is measured by its \emph{Gaschütz rank}, which we bound for all connected Lie groups, and compute exactly in most cases. Additionally, we compute the maximal size of an irredundant generating set of connected abelian Lie groups, and discuss connections between such generation problems with the Wiegold conjecture.

math.GR↗

arXiv2025

Structural properties of reduced $C^*$-algebras associated with higher-rank lattices

We present the first examples of higher-rank lattices whose reduced $C^{*}$-algebras satisfy strict comparison, stable rank one, selflessness, uniqueness of embeddings of the Jiang--Su algebra, and allow explicit computations of the Cuntz semigroup. This resolves a question raised in recent groundbreaking work of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, in which they exhibited a large class of finitely generated non-amenable groups satisfying these properties. Our proof relies on quantitative estimates in projective dynamics, crucially using the exponential mixing for diagonalizable flows. As a result, we obtain an effective mixed-identity-freeness property, which, combined with V. Lafforgue's rapid decay theorem, yields the desired conclusions.

math.OA↗

arXiv2025

Characters of diagonal products and Hilbert-Schmidt stability

We initiate a quantitative study of Hilbert-Schmidt stability for infinitely presented groups through the novel notion of stability radius growth. We exhibit an uncountable family of Hilbert-Schmidt stable amenable groups with arbitrarily large such growth. In particular, this answers a question of Lubotzky. Our approach is based on the character-theoretic stability criterion of Hadwin and Shulman. We classify the characters of alternating and elementary enrichments as well as diagonal products, including the classical family of B.H. Neumann groups.

math.GR↗

arXiv2024

Trace spaces of full free product $C^*$-algebras

We study the space of traces associated with arbitrary full free products of unital, separable $C^*$-algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the same object: the Poulsen simplex, that is, the unique infinite-dimensional metrizable Choquet simplex whose extreme points are dense. Moreover, we show that whenever such a trace space is the Poulsen simplex, the extreme points are dense in the Wasserstein topology. Concretely for the case of groups, we find that, unless the trivial character is isolated in the space of characters, the space of traces of any free product of non-trivial countable groups is the Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_{1},M_{2})$ of a tracial von Neumann algebra $M$, providing necessary conditions under which $M_{1}$ and a small unitary perturbation of $M_{2}$ generate a II$_{1}$ factor.

math.OA↗

arXiv2024

Spectral gap and character limits in arithmetic groups

We establish vanishing results for limits of characters in various discrete groups, most notably irreducible lattices in higher rank semisimple Lie groups. As an application, we show that any sequence of finite-dimensional representations converges to the regular representation in the Fell topology. We achieve this by studying the geometry of the simplex of traces of discrete groups having Kazhdan's property (T) or its relative generalizations.

math.GR↗

arXiv2024

The Space of Traces of the Free Group and Free Products of Matrix Algebras

We show that the space of traces of the free group $F_d$ on $2\leq d \leq \infty $ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form $C(X_1)*C(X_2)$ where $X_1$ and $X_2$ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n(\mathbb{C}) * M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.

math.GR↗

arXiv2023

Characters of solvable groups, Hilbert-Schmidt stability and dense periodic measures

We study the character theory of metabelian and polycyclic groups. It is used to investigate Hilbert-Schmidt stability via the character-theoretic criterion of Hadwin and Shulman. There is a close connection between stability and dynamics of automorphisms of compact abelian groups. Relying on this, we deduce that finitely generated virtually nilpotent groups, free metabelian groups, lamplighter groups as well as upper triangular groups over certain rings of algebraic integers are Hilbert-Schmidt stable.

math.GR↗

arXiv2022

Charmenability and Stiffness of Arithmetic Groups

We characterize charmenability among arithmetic groups and deduce dichotomy statements pertaining normal subgroups, characters, dynamics, representations and associated operator algebras. We do this by studying the stationary dynamics on the space of characters of the amenable radical, and in particular we establish stiffness: any stationary probability measure is invariant. This generalizes a classical result of Furstenberg for dynamics on the torus. Under a higher rank assumption, we show that any action on the space of characters of a finitely generated virtually nilpotent group is stiff.

math.GR↗

arXiv2018

On the Complexity of Two Dimensional Commuting Local Hamiltonians

The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice but this problem too remained open, apart from some very specific cases. Here we consider a wide class of two dimensional CLH instances; these are $k$-local CLHs, for any constant $k$; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [2011] which regarded the special case of $4$-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.

quant-ph↗