On the Haagerup inequality and groups acting on $\tilde{A}_n$-buildings

Alain Valette

Displaying similar documents to “On the Haagerup inequality and groups acting on ${\tilde{A}}_{n}$ -buildings”

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

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We show that there exists a finitely generated group of growth $\sim f$ for all functions $f : ℝ_{+} \to ℝ_{+}$ satisfying $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ for all $R$ large enough and $η_{+} \approx 2.4675$ the positive root of $X^{3} - X^{2} - 2 X - 4$ . Set $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; then all functions that grow uniformly faster than $exp (R^{α_{-}})$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp (R^{α})$ for a dense set of $α \in [α_{-}, 1]$ .

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

On unit group of finite semisimple group algebras of non-metabelian groups up to order 72

Gaurav Mittal, Rajendra Kumar Sharma (2021)

Mathematica Bohemica

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We characterize the unit group of semisimple group algebras $𝔽_{q} G$ of some non-metabelian groups, where $F_{q}$ is a field with $q = p^{k}$ elements for $p$ prime and a positive integer $k$ . In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_{3} \times C_{3}) ⋊ C_{3}) ⋊ C_{2}$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.

Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

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Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then ${(U^{n})}_{n \in ℕ}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $\frac{μ (U^{n + 1} ∖ U^{n})}{μ (U^{n})} .$ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain...

Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

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Let $2 \leq a \leq b \leq c \in ℕ$ with $μ = 1 / a + 1 / b + 1 / c < 1$ and let $T = T_{a, b, c} = 〈 x, y, z : x^{a} = y^{b} = z^{c} = x y z = 1 〉$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$ ? (Classically, for $(a, b, c) = (2, 3, 7)$ and more recently also for general $(a, b, c)$ .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially...

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

A note on normal generation and generation of groups

Andreas Thom (2015)

Communications in Mathematics

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In this note we study sets of normal generators of finitely presented residually $p$ -finite groups. We show that if an infinite, finitely presented, residually $p$ -finite group $G$ is normally generated by $g_{1}, \dots, g_{k}$ with order $n_{1}, \dots, n_{k} \in {1, 2, \dots} \cup {\infty}$ , then $β_{1}^{(2)} (G) \leq k - 1 - \sum_{i = 1}^{k} \frac{1}{n_{i}},$ where $β_{1}^{(2)} (G)$ denotes the first $ℓ^{2}$ -Betti number of $G$ . We also show that any $k$ -generated group with $β_{1}^{(2)} (G) \geq k - 1 - ε$ must have girth greater than or equal $1 / ε$ .

On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Supercritical self-avoiding walks are space-filling

Hugo Duminil-Copin, Gady Kozma, Ariel Yadin (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $γ$ between two points on the boundary of a finite subdomain of $ℤ^{d}$ is proportional to $μ^{- length (γ)}$ . When $μ$ is supercritical (i.e. $μ l t; μ_{c}$ where $μ_{c}$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

Expansion in $S L_{d} (𝒪_{K} / I)$ , $I$ square-free

Péter P. Varjú (2012)

Journal of the European Mathematical Society

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Let $S$ be a fixed symmetric finite subset of $S L_{d} (𝒪_{K})$ that generates a Zariski dense subgroup of $S L_{d} (𝒪_{K})$ when we consider it as an algebraic group over $m a t h b b Q$ by restriction of scalars. We prove that the Cayley graphs of $S L_{d} (𝒪_{K} / I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of $𝒪_{K}$ if $d = 2$ and $K$ is an arbitrary numberfield, or if $d = 3$ and $K = ℚ$ .

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p = tp (b / A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd (A)$ , (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ ...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder, A. Veselov (2005)

Journal of the European Mathematical Society

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We consider both standard and twisted actions of a (real) Coxeter group $G$ on the complement $ℳ_{G}$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{*} (ℳ_{G}, 𝒞)$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_{n}$ , the...

Displaying similar documents to “On the Haagerup inequality and groups acting on A ˜ n -buildings”

Displaying similar documents to “On the Haagerup inequality and groups acting on ${\tilde{A}}_{n}$ -buildings”