Adrian Ioana
We construct a sofic approximation of $\mathbb F_2\times\mathbb F_2$ that is not essentially a "branched cover" of a sofic approximation by homomorphisms. This answers a question of L. Bowen.
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Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more general metric approximations of groups under wreath products. Our main result is the following. Suppose that $H$ is a sofic group and $G$ is a countable, discrete group. If $G$ is sofic, hyperlinear, weakly sofic, or linear sofic, then $G\wr H$ is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric ap…
Given sofic approximations for countable, discrete groups $G,H$, we construct a sofic approximation for their wreath product $G\wr H$.
Hyperlinear approximations to amenable groups come from sofic approximations
本文定量证明了顺从群的每个超线性近似均可由合适的sofic近似模拟，利用高维球面测度集中控制算子矩阵系数偏差，并得出sofic与超线性近似稳定性之间的联系。
On sofic approximations of non amenable groups
本文对每个初始次顺从的非顺从群构造了两个在任何万有sofic群自同构下均不共轭的sofic逼近，回答了Paunescu的问题并推广了Elek-Szabo唯一性定理。
Sofic boundaries of groups and coarse geometry of sofic approximations
研究通过索菲近似定义群的索菲边界，证明大尺度性质如属性A、Hilbert空间渐近嵌入性和几何属性(T)对应于群的分析性质，扩展了残余有限群的研究成果。
Hyperlinear approximations to amenable groups come from sofic approximations
本文给出了可遗群双线性性与sofic性等价的定量表述，证明双线性近似源于sofic近似，并建立了Hilbert-Schmidt与置换稳定性间的定量关系。
We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations solvable in all alternating groups is solvable over $G$. The last characterization allows to express soficity of an existentially closed group by $\forall\exists$-sentences. Keywords: sofic groups, approximations, equations over groups.
A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag--Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. We construct variations on Higman's group replacing the Baumslag--Solitar group by other groups $G$. We give an elementary condition on $G$, enjoyed for example by $\mathbb{Z} \wr \mathbb{Z}$ and the integral Heisenberg group, under which the resulting group is sofic. …
Approximate homomorphisms and sofic approximations of orbit equivalence relations
本文证明可数群的近似同态序列可实现为轨道等价关系的sofic逼近的限制，并由不变随机子群唯一确定，进而恢复了顺从群近似同态的已知稳定性与共轭刻画。
Centralizers of sofic approximations of Kazhdan groups
该研究证明了满足特定软嵌入条件的Kazhdan群局部可嵌入有限群，有限呈现的此类群为剩余有限，并得出其软嵌入的中心化子是置换群度量超积的结论。
We prove the algebraic eigenvalue conjecture of J. Dodziuk, P. Linnell, V. Mathai, T. Schick and S. Yates for sofic groups. Moreover, we give restrictions on the spectral measure of elements in the integral group ring. Finally, we define integer operators and prove a quantization of the operator norm below 2.
Geometric property (T) for box spaces and sofic approximations
该研究证明了箱空间和索菲克近似中几何性质（T）的等价性，并提出了基于谱间隙的图序列近似同构的条件。
We give a simple and unified proof showing that the unrestricted wreath product of a weakly sofic, sofic, linear sofic, or hyperlinear group by an amenable group is weakly sofic, sofic, linear sofic, or hyperlinear, respectively. By means of the Kaloujnine-Krasner theorem, this implies that group extensions with amenable quotients preserve the four aforementioned metric approximation properties. We also discuss the case of co-amenable groups.
Sofic actions, halo products, and metric approximations of groups
本文提出sofic C-作用的概念，展示其在构造新群类及统一现有结果中的作用。
We show that for every sofic chunk $E$ there is a bijective homomorphism $f: E_c \rightarrow E$, where $E_c$ is a chunk of the group of computable permutations of $\mathbb{N}$ so that the approximating morphisms of $E$ can be viewed as restrictions of permutations of $E_c$ to finite subsets of $\mathbb{N}$. Using this we study some relevant effectivity conditions associated with sofic chunks and their profiles.
This is a paper with two aims. First, we show that the map from $\mathbb{Z}/p\mathbb{Z}$ to itself defined by exponentiation $x\to m^x$ has few $3$-cycles -- that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds. Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\mathbb{Z}/p\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property.
Measure equivalence and sofic approximations
本文提出一种利用sofic逼近间的映射产生sofic群测度耦合的技术，证明了可解Baumslag-Solitar群、Lamplighters群和SOL群彼此指数测度等价，这与它们通常不拟等距形成鲜明对比。
We undertake a systematic study of the approximation properties of the topological and measurable versions of the coarse boundary groupoid associated to a sequence of finite graphs of bounded degree. On the topological side, we prove that asymptotic coarse embeddability of the graph sequence into a Hilbert space is equivalent to the coarse boundary groupoid being topologically a-T-menable, thus answering a question by Rufus Willett. On the measure-theoretic side, we prove that measure-theoretic amenability resp. a-T-menability of the coarse boundary groupoid are related to hyperfiniteness and …
Sofic approximations and quantitative measure couplings
通过 Brieussel-Zheng 对角积到灯夫群的构造，建立定量测度子群耦合，并用于解决灯夫群情形下给定群与给定量化的逆问题。
We prove Bowen's conjecture that every sequence of finite graphs that locally converges to the Cayley graph of a countably infinite group with Kazhdan Property (T) is essentially a vertex-disjoint union of expander graphs. We characterize graph sequences that are essentially a vertex-disjoint union of expander graphs in terms of the Markov operator.
Optimal Linear Sofic Approximations Of Countable Groups
本文研究可数群的最优线性sofic逼近，证明每个线性sofic群都是1/2-线性sofic的且该常数无法改进，无挠群则达到1-线性sofic，并提供了有限群在复数和正特征域上的最优常数及证明所需的随机游走非集中估计。
High-dimensional expansion and soficity of groups
本文构造了特定格，证明其泛中心扩张在弱稳定性下非 sofic，并利用高维扩张构造了任意有限阿贝尔群上的新余环扩张子。
A sofic group $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if $\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for some $d \geq 5$ then there exists a group which is not sofic.
The Lanford-Ruelle theorem for actions of sofic groups
本研究将经典Lanford-Ruelle定理推广至sofic群作用情形，证明满足拓扑Markov性质的子移位上，最大化sofic压强的平移不变测度为Gibbs态，还给出平衡测度唯一性等应用及Chung变分原理的简短证明。
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 r…
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…
Let $G$ be a group. Let $X$ be a connected algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of $K$-points of $X$. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over $(G,X,K)$. They are cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M\subset G$ is a finite memory set of $τ$. Our first result is that when $G$ is sofic, such an algebraic group cellular automaton $τ$ is invertible whenever it is injective and $\text{char}(K)=0…
This thesis aims to serve as an introduction to the theory of quasitilings for amenable groups. In order to showcase the power of this theory, we focus on the study of the Sofic Lück Approximation Conjecture, which can be proven for amenable groups by making use of quasitilings. The first four chapters of the thesis are an exposition of the aforementioned topics, collected from the literature. After that, in the fifth and final chapter we present some new results related to the Sofic Lück Approximation Conjecture for amenable groups over discrete valuation rings and number fields.
Inapproximability of actions and Kazhdan's property (T)
该研究针对具有Kazhdan性质(T)的可数群G与有限生成群D，通过sofic逼近的扩张子图性质，推导D的LEF性，明确非LEF时G×D作用的极限性质，并给出几乎自同构的自改进定理作为关键技术支撑。
We prove that the topological full group $[[X]]$ of a two-sided full shift $X = Σ^{\mathbb{Z}}$ contains every right-angled Artin group (also called a graph group). More generally, we show that the family of subgroups with "linear look-ahead" is closed under graph products. We show that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ embeds in $[[X]]$, and conjecture that it does not embed in $[[X]]$ with linear look-ahead. Generalizing the lamplighter group, we show that whenever $G$ acts with "unique moves" (or at least "move-$A$ithfully"), we have $A \wr G \leq [[X]]$ for finite abelian…
The super approximation property of $\mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z}) \times \mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z})$
本文证明了在 $\mathrm{SL}_2(\mathbb{Z})^3$ 中由有限对称集生成的 Zariski-稠密群，其模 $q$ 的 Cayley 图族构成 expander，从而确立了该群的超逼近性质。
Let $Γ_2\subseteq Γ_1$ be finitely generated subgroups of ${\rm GL}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let $\mathbb{G}_i$ be the Zariski-closure of $Γ_i$ in $({\rm GL}_{n_0})_{\mathbb{Q}}$, $\mathbb{G}_i^{\circ}$ be the Zariski-connected component of $\mathbb{G}_i$, and let $G_i$ be the closure of $Γ_i$ in $\prod_{p\nmid q_0}{\rm GL}_{n_0}(\mathbb{Z}_p)$. In this article we prove that, if $\mathbb{G}_1^{\circ}$ is the smallest closed normal subgroup of $\mathbb{G}_1^{\circ}$ which contains $\mathbb{G}_2^{\circ}$ and $Γ_2\curvearrowright G_2$ has spectral gap, then $Γ_1\curvearrowrigh…
Super approximation for $\text{SL}_2\times \text{SL}_2$ and $\text{ASL}_2$
研究SL₂×SL₂和ASL₂中有限对称集生成的群的Cayley图展开性质，证明其为展开器家族。
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