Uzy Hadad
We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of $p$-groups of lower $p$-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.
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Kazhdan constants for two-element generating sets of $\mathrm{SL}_n(\mathbb Z)$
本文证明对每个$n\geq3$，$\mathrm{SL}_n(\mathbb Z)$的所有两元素生成集的Kazhdan常数下确界为$0$，回答了生成集基数有界时下确界是否仍为零的问题。
In this article we improve the known Kazhdan constant for $SL_n(Z)$ with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by $[42\sqrt{n}+860]^{-1}$, which gives the exact asymptotic behavior of the Kazhdan constant, as $n$ goes to infinity, since $\sqrt{2/n}$ is an upper bound. We can use this bound to improve the bounds for the spectral gap of the Cayley graph of $SL_n(F_p)$ and for the working time of the product replacement algorithm for abelian groups.
Let $R$ be a ring generated by $l$ elements with stable range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant $ε_0>0$ for some $d > r $. We prove that there exist $ε(ε_0,l) >0$ and $k \in N$, s.t. for every $n \geq d$, $EL_n(R)$ has a generating set of order $k$ and a Kazhdan constant larger than $ε$. As a consequence, we obtain for $SL_n(Z)$ where $n \geq 3$, a Kazhdan constant which is independent of $n$ w.r.t generating set of a fixed size.
Small Kazhdan constants for small generating sets of linear groups
本文证明线性群及具有非初等收敛作用的有限生成群，存在大小仅比生成元数多2的生成集，其Kazhdan常数可任意小。
We prove that for any finite index subgroup $\Ga$ in $SL_n(\mathbb{Z})$, there exists $k=k(n)\in\mathbb{N}$, $\ep=\ep(\Ga)>0$, and an infinite family of finite index subgroups in $\Ga$ with a Kazhdan constant greater than $\ep$ with respect to a generating set of order $k$. On the other hand, we prove that for any finite index subgroup $\Ga$ of $SL_n(\mathbb{Z})$, and for any $\ep>0$ and $k\in \mathbb{N}$, there exists a finite index subgroup $\Ga'\leq \Ga$ such that the Kazhdan constant of any finite index subgroup in $\Ga'$ is less than $\ep$, with respect to any generating set of order $k$.…
Kazhdan constants of discrete groups are hard to compute and the actual constants are known only for several classes of groups. By solving a semidefinite programming problem by a computer, we obtain a lower bound of the Kazhdan constant of a discrete group. Positive lower bounds imply that the group has property (T). We study lattices on $\tilde{A}_2$-buildings in detail. For $\tilde{A}_2$-groups, our numerical bounds look identical to the known actual constants. That suggests that our approach is effective. For a family of groups, $G_1, \cdots, G_4$, that are studied by Ronan, Tits and others…
We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.
Let $G=\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ and $H=\mathrm{SL}(2,\mathbb{Z})$. We prove that the action $G\curvearrowright\mathbb{R}^2$ is uniformly non-amenable and that the quasi-regular representation of $G$ on $\ell^2(G/H)$ has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.
Non Uniform Kazhdan Constant for Linear Groups
该研究证明，存在任意多非平凡有限群直积商群的有限生成群，其一致Kazhdan常数为0，并将此结果应用于SL_n(ℤ)，得出其一致Kazhdan常数为0的结论。
Kazhdan constants for Chevalley groups over the integers
本文研究整数环上Chevalley群的Kazhdan常数下界问题，通过建立根系结构与Laplace算子平方行为的新联系，得到了除$\u27e8\mathtt{A}_{n}\u27e9$外类型的首个显式渐近精确下界。
A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real line
研究了实数线上同胚群的Lipschitz和Kazhdan常数之间的联系，得出了相关群作用的Lipschitz和Kazhdan常数的上下界。
Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\dots,p_{r}$ there exists a continuous probability measure $μ$ on the unit circle $\mathbb{T}$ such that \[ \inf_{k_{1}\ge 0,\dots,k_{r}\ge 0}|\widehat{μ}(p_{1}^{k_{1}}\dots p_{r}^{k_{r}})|>0. \] This results applies in particular to the Furstenberg set $F=\{2^{k}3^{k'}\,;\,k\ge 0,\ k'\ge 0\}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $\times 2$-$\times 3$ conjecture. We also estimate the modified Kazhdan constant of…
We reformulate and extend the geometric method for proving Kazhdan property T developed by Dymara and Januszkiewicz and used by Ershov and Jaikin. The main result says that a group G, generated by finite subgroups G_i, has property T if the group generated by each pair of subgroups has property T and sufficiently large Kazhdan constant. Essentially, the same result was proven by Dymara and Januszkiewicz, however our bound for "sufficiently large" is significantly better. As an application of this result, we give exact bounds for the Kazhdan constants and the spectral gaps of the random walks o…
In 2010, Invent. Math., Ershov and Jaikin-Zapirain proved Kazhdan's property (T) for elementary groups. This expository article focuses on presenting an alternative simpler proof of that. Unlike the original one, our proof supplies no estimate of Kazhdan constants. It may be regarded as a specific example of the results in the paper "Upgrading fixed points without bounded generation" (arXiv:1505.06728, forthcoming version) by the author.
It is well-known that a finitely generated group $Γ$ has Kazhdan's property (T) if and only if the Laplacian element $Δ$ in ${\mathbb R}[Γ]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in ${\mathbb R}[Γ]$. Namely, $Γ$ has property (T) if and only if there are a constant $κ>0$ and a finite sequence $ξ_1,...,ξ_n$ in ${\mathbb R}[Γ]$ such that $Δ^2-κΔ= \sum_i ξ_i^*ξ_i$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in ${\mathbb R}[Γ]$, possibly with an assist of computers.
Let Γ(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. Γ(n,p) is free and if p >= C log n/n^2 then a.a.s. Γ(n,p) has Kazhdan's property (T). Furthermore, we show that there exist constants C',c' > 0 such that if C'/n^2 <= p <= c' log n/n^2, then a.a.s. Γ(n,p) is neither free nor has Kazhdan's property (T).
In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups $U_ζ$ (at roots of unity $ζ$). These bounds depend only on the root system, and not on the characteristic $p$ or the size of the root of unity $ζ$. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system $Φ$. For …
We prove that $\operatorname{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$. We also provide explicit lower bounds for the Kazhdan constants of $\operatorname{SAut}(F_n)$ (with $n \geqslant 6$) and of $\operatorname{SL}_n(\mathbb{Z})$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.
This paper consists of three interconnected parts. Parts I,III study the relationship between the cohomology of a reductive group and that of a Levi subgroup. For example, we provide a necessary condition, arising from Kazhdan-Lusztig theory, for the natural map on Ext-groups of irreducible modules to be surjective. In cohomological degree 1, the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer obtained for general linear groups, and they both extend and improve upon his work when our condition is met. Part II obtains results on Lusztig c…
Discrete group extensions of countable Markov shifts and pressure gap
本文将关于顺从性和增长间隙的若干经典结果推广到强正递归的可数马尔可夫移位，利用扭曲Patterson--Sullivan测度的方法，并给出扭曲测度的背景及一个例子的估计。
We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $βr^δ$ for some explicit constants $β, δ> 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a nove…
We consider the Iwahori-Hecke algebra associated to an almost split Kac-Moody group $G$ (affine or not) over a nonarchimedean local field $K$. It has a canonical double-coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ indexed by a sub-semigroup $W^+$ of the affine Weyl group $W$. The multiplication is given by structure constants $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in N=Z_{\geq0}$ : $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$. A conjecture, by Bravermann, Kazhdan, Patnaik, Gaussent and the authors, tells that $a…
L. Babai has shown that a faithful permutation representation of a nonsplit extension of a group by an alternating group $A_k$ must have degree at least $k^2(\frac{1}{2}-o(1))$, and has asked how sharp this lower bound is. We prove that Babai's bound is sharp (up to a constant factor), by showing that there are such nonsplit extensions that have faithful permutation representations of degree $\frac{3}{2}k(k-1)$. We also reprove Babai's quadratic lower bound with the constant $\frac{1}{2}$ improved to 1 (by completely different methods).
Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show that this occurs if and only if d < 5. Secondly, we consider the question of whether there exists a constant c such that if K/k is a finite subextension of k^[d], then K is generated over k by elements of degree at most c. This was previously considered by Checcoli. We show that such a constant exists if and only if d < 3. This question becomes more interestin…
Central extensions and almost representations
本文利用 de la Harpe-Skandalis 预行列式构造酉群的典范中心扩张，将渐近群同态的阻碍类刻画为 2-上同调类，并由此得到包含绕数不变量在内的数值不变量，进而证明非零实 2-上同调会阻碍离散群全群 $C^*$-代数的 $C^*$-稳定性。
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