arXiv subjects

Goulnara Arzhantseva

Publications and source records attributed to Goulnara Arzhantseva.

At least 19 recordsLinked to original sources

arXiv2026

Lacunary hyperbolic groups with fast injectivity radius growth and enough loxodromic elements are selfless

We prove that a lacunary hyperbolic group $G = \varinjlim G_i$ with sufficient generics is selfless in the sense of Amrutam--Gao--Kunnawalkam Elayavalli--Patchell, provided the hyperbolicity constants $δ_i$ and injectivity radii $r_i$ satisfy $δ_i(\log r_i)^{7} = o(r_i)$. The proof replaces the acylindricity-based machinery of that work with a direct geodesic $n$-gon criterion due to Arzhantseva, which applies in any $δ$-hyperbolic space. As a consequence, combined with rapid decay, $G$ is $C^*$-selfless. The condition is mild: torsion-free Tarski monsters, Jacobson's mixed-identity-free elementary amenable groups and Gromov monster groups satisfy it for appropriate parameter choices. The amenable examples are selfless but cannot be $C^*$-selfless, providing examples that separate these properties. Finally we remark that the Gromov monster group examples provide a potential avenue to a non-exact $C^*$-algebra with strict comparison.

math.GR↗

arXiv2024

Constraint stability in permutations and action traces

An action trace is a function naturally associated to a probability measure preserving action of a group on a standard probability space. For countable amenable groups, we characterise stability in permutations using action traces. We extend such a characterisation to constraint stability. We give sufficient conditions for a group to be constraint stable. As an application, we obtain many new examples of groups stable in permutations, in particular, among free amalgamated products over a finite group. This is the first general result (besides trivial case of free products) which gives a wealth of non-amenable groups stable in permutations.

math.GR↗

arXiv2024

Spectral gap and origami expanders

We construct the first measure-preserving affine actions with spectral gap on surfaces of arbitrary genus $g > 1$. We achieve this by finding geometric representatives of multi-twists on origami surfaces. As a major application, we construct new expanders that are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. Our methods also show that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander.

math.MG↗

arXiv2022

Logarithmic girth expander graphs of $SL_n(\mathbb F_p)$

We provide an explicit construction of finite 4-regular graphs $(Γ_k)_{k\in \mathbb N}$ with ${girth Γ_k\to\infty}$ as $k\to\infty$ and $\frac{diam Γ_k}{girth Γ_k}\leqslant D$ for some $D>0$ and all $k\in\mathbb{N}$. For each fixed dimension $n\geqslant 2,$ we find a pair of matrices in $SL_{n}(\mathbb{Z})$ such that (i) they generate a free subgroup, (ii)~their reductions $\bmod\, p$ generate $SL_{n}(\mathbb{F}_{p})$ for all sufficiently large primes $p$, (iii) the corresponding Cayley graphs of $SL_{n}(\mathbb{F}_{p})$ have girth at least $c_n\log p$ for some $c_n>0$. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most $O(\log p)$. This gives infinite sequences of finite $4$-regular Cayley graphs of $SL_n(\mathbb F_p)$ as $p\to\infty$ with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions $n\geqslant 2$ (all prior examples were in $n=2$). Moreover, they happen to be expanders. Together with Margulis' and Lubotzky-Phillips-Sarnak's classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders.

math.GR↗

arXiv2020

Quantifying metric approximations of discrete groups

We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_α, d_α, k_α, \varepsilon_{α})\}_{α\in I, }$ where each group $G_α$ is equipped with a bi-invariant metric $d_α$ and a dimension $k_α$, for strictly positive real numbers $\varepsilon_{α}$ such that $\inf_{α}\varepsilon_{α}>0$. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Folner) profiles of amenable groups and recently introduced functions quantifying residually finite groups. Our viewpoint is much more general and covers hyperlinear and sofic approximations as well as many other metric approximations such as weakly sofic, weakly hyperlinear, and linear sofic approximations.

math.GR↗

arXiv2020

On approximation properties of semidirect products of groups

Let R be a class of groups closed under taking semidirect products with finite kernel and fully residually R-groups. We prove that R contains all R-by-{finitely generated residually finite} groups. It follows that a semidirect product of a finitely generated residually finite group with a surjunctive group is surjunctive. This remained unknown even for direct products of a surjunctive group with the integers Z.

math.GR↗

arXiv2018

Unrestricted wreath products and sofic groups

We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argument, to hyperlinear-by-amenable groups.

math.GR↗

arXiv2017

Admitting a coarse embedding is not preserved under group extensions

We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.

math.GR↗

arXiv2017

Constraint metric approximations and equations in groups

We introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation. This is done in the language of group equations with coefficients. We give an example of a group which is not constraintly sofic. In building it, we find a sofic representation of free group with trivial commutant among extreme points of the convex structure on the space of sofic representations. We consider the centralizer equation in permutations as an instance of this new general setting. We characterize permutations $p\in S_k$ whose centralizer is stable in permutations with respect to the normalized Hamming distance, that is, a permutation which almost centralizes $p$ is near a centralizing permutation. This answers a question of Gorenstein-Sandler-Mills (1962).

math.GR↗

arXiv2017

A characterization for asymptotic dimension growth

We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. We also apply our new characterization to geodesic coarse median spaces of finite rank and establish that they have subexponential asymptotic dimension growth. This strengthens a recent result of J. Spakula and N. Wright.

math.MG↗

arXiv2016

Relative expanders

We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n\trianglelefteq G$ such that for every finite generating subset $S\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does not coarsely embed into any $L^p$-space for $1\leqslant p<\infty$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander. The reason why our examples do not coarsely embed is a new phenomenon called relative expansion, which we define in terms of Poincaré inequalities.

math.GR↗

arXiv2015

Almost commuting permutations are near commuting permutations

We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations. Our result extends to $k$-tuples of almost commuting permutations, for any given $k$, and allows restrictions, for instance, to even permutations.

math.GR↗

arXiv2015

Infinitely presented small cancellation groups have the Haagerup property

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the C'(1/6)-small cancellation condition. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum-Connes conjecture holds for them. The result is a first non-trivial advancement in understanding groups with such properties among infinitely presented non-amenable direct limits of hyperbolic groups. The proof uses the structure of a space with walls introduced by Wise. As the main step we show that C'(1/6)-complexes satisfy the linear separation property.

math.GR↗

arXiv2014

Graphical small cancellation groups with the Haagerup property

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the graphical C'(lambda)-small cancellation condition with respect to graphs endowed with a compatible wall structure. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum-Connes conjecture and, hence, the Baum-Connes conjecture with arbitrary coefficients hold for them. As the main step we show that C'(lambda)-complexes satisfy the linear separation property. Our result provides many new examples and a general technique to show the Haagerup property for graphical small cancellation groups.

math.GR↗

arXiv2014

Rips construction without unique product

Given a finitely presented group $Q,$ we produce a short exact sequence $1\to N \hookrightarrow G \twoheadrightarrow Q \to 1$ such that $G$ is a torsion-free Gromov hyperbolic group without the unique product property and $N$ is without the unique product property and has Kazhdan's Property (T). Varying $Q,$ we show a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. As an application, we obtain Tarski monster groups without the unique product property.

math.GR↗

arXiv2012

Linear sofic groups and algebras

We introduce and systematically study linear sofic groups and linear sofic algebras. This generalizes amenable and LEF groups and algebras. We prove that a group is linear sofic if and only if its group algebra is linear sofic. We show that linear soficity for groups is a priori weaker than soficity but stronger than weak soficity. We also provide an alternative proof of a result of Elek and Szabo which states that sofic groups satisfy Kaplansky's direct finiteness conjecture.

math.GR↗

arXiv2012

Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power continuum. The present work can be viewed as a first non-trivial step towards a systematic investigation of direct limits of hyperbolic groups.

math.GR↗

arXiv2012

Isomorphism versus commensurability for a class of finitely presented groups

We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem.

math.GR↗