scholarly journals Dynamical properties of profinite actions

2011 ◽  
Vol 32 (6) ◽  
pp. 1805-1835 ◽  
Author(s):  
MIKLÓS ABÉRT ◽  
GÁBOR ELEK
Keyword(s):  
Finite Index ◽  
Linear Groups ◽  
Regular Graphs ◽  
Weak Containment ◽  
Residually Finite Groups ◽  
The Family ◽  
Property T ◽  
Free Actions ◽  
Theoretical Consequence ◽  
Lattice Of Subgroups

AbstractWe study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

scholarly journals Complete Bredon cohomology and its applications to hierarchically defined groups

2016 ◽  
Vol 161 (1) ◽  
pp. 143-156
Author(s):  
BRITA E. A. NUCINKIS ◽  
NANSEN PETROSYAN
Keyword(s):  
Integral Characteristic ◽  
Linear Groups ◽  
Dimensional Model ◽  
Classifying Space ◽  
Soluble Groups ◽  
3 Dimensional ◽  
Cyclic Groups ◽  
Bredon Cohomology ◽  
Finite Dimensional ◽  
The Family

AbstractBy considering the Bredon analogue of complete cohomology of a group, we show that every group in the class$\cll\clh^{\mathfrak F}{\mathfrak F}$of type Bredon-FP∞admits a finite dimensional model for$E_{\frak F}G$.We also show that abelian-by-infinite cyclic groups admit a 3-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for$\mathfrak {F}$, the class of virtually cyclic groups, the class of$\cll\clh^{\mathfrak F}{\mathfrak F}$-groups contains all locally virtually soluble groups and all linear groups over${\mathbb{C}}$of integral characteristic.

scholarly journals Free actions of free groups on countable structures and property (T)

2016 ◽  
Vol 232 (1) ◽  
pp. 49-63 ◽  
Author(s):  
David M. Evans ◽  
Todor Tsankov
Keyword(s):  
Free Groups ◽  
Property T ◽  
Free Actions

scholarly journals Parallel Context-Free Languages

1974 ◽  
Vol 3 (30) ◽  
Author(s):  
Sven Skyum
Keyword(s):  
Finite Index ◽  
The Family ◽  
And Control ◽  
Context Free

<p>The relation between the family of context-free languages and the family of parallel context-free languages is examined in this paper. It is proved that the families are incomparable. Finally we prove that the family of languages of finite index is contained in the family of parallel context-free languages.</p><p>Information and Control, 26 (1974) pp. 280-285.</p>

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

scholarly journals PROJECTIVE SPECIAL LINEAR GROUPS PSL4(q) ARE DETERMINED BY THE SET OF THEIR CHARACTER DEGREES

2012 ◽  
Vol 11 (06) ◽  
pp. 1250108 ◽  
Author(s):  
HUNG NGOC NGUYEN ◽  
HUNG P. TONG-VIET ◽  
THOMAS P. WAKEFIELD
Keyword(s):  
Finite Group ◽  
Character Degrees ◽  
Linear Groups ◽  
Complex Character ◽  
Special Linear ◽  
Projective Special Linear Groups ◽  
Special Linear Groups ◽  
The Family ◽  
Group A ◽  
A Finite Group

Let G be a finite group and let cd (G) be the set of all irreducible complex character degrees of G. It was conjectured by Huppert in Illinois J. Math.44 (2000) that, for every non-abelian finite simple group H, if cd (G) = cd (H) then G ≅ H × A for some abelian group A. In this paper, we confirm the conjecture for the family of projective special linear groups PSL 4(q) with q ≥ 13.

scholarly journals On the Number of Forests and Connected Spanning Subgraphs

2021 ◽  
Author(s):  
Márton Borbényi ◽  
Péter Csikvári ◽  
Haoran Luo
Keyword(s):  
Connected Graph ◽  
Average Degree ◽  
Regular Graphs ◽  
Spanning Subgraphs ◽  
The Family

AbstractLet F(G) be the number of forests of a graph G. Similarly let C(G) be the number of connected spanning subgraphs of a connected graph G. We bound F(G) and C(G) for regular graphs and for graphs with a fixed average degree. Among many other things we study $$f_d=\sup _{G\in {\mathcal {G}}_d}F(G)^{1/v(G)}$$ f d = sup G ∈ G d F ( G ) 1 / v ( G ) , where $${\mathcal {G}}_d$$ G d is the family of d-regular graphs, and v(G) denotes the number of vertices of a graph G. We show that $$f_3=2^{3/2}$$ f 3 = 2 3 / 2 , and if $$(G_n)_n$$ ( G n ) n is a sequence of 3-regular graphs with the length of the shortest cycle tending to infinity, then $$\lim _{n\rightarrow \infty }F(G_n)^{1/v(G_n)}=2^{3/2}$$ lim n → ∞ F ( G n ) 1 / v ( G n ) = 2 3 / 2 . We also improve on the previous best bounds on $$f_d$$ f d for $$4\le d\le 9$$ 4 ≤ d ≤ 9 .

scholarly journals On small world non-Sunada twins and cellular Voronoi diagrams

2020 ◽  
Vol 30 (1) ◽  
pp. 118-142
Author(s):  
V. Ustimenko ◽  
Keyword(s):  
Voronoi Diagrams ◽  
Small World ◽  
Regular Graphs ◽  
Explicit Constructions ◽  
The Family ◽  
Small World Graphs ◽  
Edge Transitive ◽  
Transitive Graphs

Special infinite families of regular graphs of unbounded degree and of bounded diameter (small world graphs) are considered. Two families of small world graphs Gi and Hi form a family of non-Sunada twins if Gi and Hi are isospectral of bounded diameter but groups Aut(Gi) and Aut(Hi) are nonisomorphic. We say that a family of non-Sunada twins is unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. If all Gi and Hi are edge-transitive we have a balanced family of small world non-Sunada twins. We say that a family of non-Sunada twins is strongly unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. We use term edge disbalanced for the family of non-Sunada twins such that all graphs Gi and Hi are edge-intransitive. We present explicit constructions of the above defined families. Two new families of distance-regular—but not distance-transitive—graphs will be introduced.

scholarly journals Strong Banach property (T) for simple algebraic groups of higher rank

2014 ◽  
Vol 06 (01) ◽  
pp. 75-105 ◽  
Author(s):  
Benben Liao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Algebraic Groups ◽  
Isometric Action ◽  
Cocompact Lattice ◽  
Uniform Embedding ◽  
The Family ◽  
Property T ◽  
Higher Rank ◽  
Finite Quotients

We extend Vincent Lafforgue's results to Sp4. As applications, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1, and any affine isometric action of such a group, or of any cocompact lattice in it, in a Banach space of type > 1 has a fixed point.

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