Construction of finite groups amalgams and geometries. Geometries of the group U4(2)

The paper describes methods of constructing of residually connected and flag-transitive geometries and corresponding presentations cf finite groups. The methods are illustrated on the finite simple group U₄(2). As a result all residually connected and flag-transitive geometries of the group U₄(2) with maximsl subgroups as the stabilizers of geometries' elements (exclsding rank 4 geometries with a trivial Borel subgroup) were described. USing these geometries several "natural" presentations of the group U₄(2) were obtained.     

On questions related to saturated chains of primes in polynomial rings

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X₁,…,X_n〉 residually Jaffard (resp., totally Jaffard) when R(X₁,…,X_n) is ? We construct a three-dimensional local ring R such that R(X₁,…,X_n) is totally Jaffard (and hence, residually Jaffard) whereas R〈X₁,…,X_n〉 is not residually Jaffard (and hence, not totally Jaffard).     

Highly symmetric hypertopes

We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytope theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and \({C^+}\)-groups in the chiral case.     

A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS ⊂ G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G → F_Ssuch thath_S(g) ≠ 1 for allg ∈ S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.     

The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups

The following theorem is proved: LetG be any group. Then the augmentation ideal ofZG is residually nilpotent if and only ifG is approximated by nilpotent groups without torsion or discriminated by nilpotent p_i,-groups,i ∈I, of finite exponents. This theorem is applied to obtain conditions under which the groupsF/N′ are residually nilpotent whereF is a free non-cyclic group and N?F.     

Geometric Property （T）

This paper discusses“geometric property （T）”. This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property （T） is a strong form of “expansion property”, in particular, for a sequence （Xn） of bounded degree finite graphs, it is strictly stronger than （Xn） being an expander in the sense that the Cheeger constants h（Xn） are bounded below. In this paper, the authors show that geometric property （T） is a coarse invariant, i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property （T） interacts with amenability, property （T） for groups, and coarse geometric notions of a-T-menability. In particular, it is shown that property （T） for a residually finite group is characterised by geometric property （T） for its finite quotients.     

Note on residually finite rings

This paper is on the subject of residually finite (= RF) modules and rings introduced by Varadarajan [93] and [98/99]. Specifically there are several theorems that simplify proofs and generalize some results of Varadarajan, namely.

Theorem 1. An RF right R-module is finitely bedded (= has finite essential socle iff M is finite.

Corollay. If T is a right RF woth just finitely many simple ringht R-modules, them R is fimite.

Theorem 2. A commutative ring R is residually finite iff every local ring R_m at a maximal ideal m is finite.     

On a Rank Five Geometry of Meixner for the Mathieu Group M12

We construct a rank five residually connected and firm geometry on which the Mathieu group M ₁₂ acts flag-transitively and residually weakly primitively (RWPRI). The group M ₁₂ is the group of automorphisms of and Aut(M ₁₂) is the correlation group of , in particular is self-dual. The diagram of is the following. Moreover satisfies the conditions (IP)₂ and (2T)₁. As a corollary, we obtain that the (RWPRI+(IP)₂)-rank of M ₁₂ is 5.     

On the length equalities for one-dimensional rings

In this article we characterize noetherian local one-dimensional analytically irreducible and residually rational domains (R,m_R) which are non-Gorenstein, the non-negative integer is equal to τ_R-1 and ?(R/(C+xR))=2, where τ_R is the Cohen-Macaulay type of R, C is the conductor of R in the integral closure of R in its quotient field Q(R) and xR is a minimal reduction of m by giving some conditions on the numerical semi-group v(R) of R.     

On Flat Flag-Transitive c.c *-Geometries

We study flat flag-transitive c.c ^*-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2²ⁿ · L _n(2) and covered by the truncated Coxeter complex of type D ₂ ⁿ . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 2²ⁿ · (2 ^n?1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.     

On the existence of certain generalized Moore geometries,part I

Using methods of Algebraic Graph Theory, generalized Moore geometries of type GM_m(s, t, c) with c = s + 1 are investigated. It is shown that such geometries do not exist for odd values of the diameter m exceeding 7, if st>1.     

Some Infinite Families of Locally Partial Geometries Generalizing the Classical Laguerre Planes

Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β 1, we construct inductively a series of rank n residually connected geometries Γ^{(n, β, Ω)}, n β, by putting Γ^{(β, β, Ω)} = Ω and extending Γ^{(n - 1, β, Ω)} with a partial geometry.     

On deformation rings of residually reducible Galois representations and R = T theorems

We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ ₀ of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ ₁ and ρ ₂. Under some assumptions on Selmer groups associated with ρ ₁ and ρ ₂ we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.     

The Groups of Motions of Some Three-Dimensional Maximal Mobility Geometries

We find the groups of motions of eight three-dimensional maximal mobility geometries. These groups are actions of just three Lie groups SL₂(R)ΔN, SL₂(C) _R , and SL₂(R)?SL₂(R) on the space R³, where N is a normal abelian subgroup. We also find explicit expressions for these actions.     

Residual properties of free groups,II

In this paper we prove that if X is an infinite class of flnite simple classical groups, then F₂, the free group of rank 2, is residually X. This solves a special case of a question of W.Magnus. He conjectures that F₂ is residually X for any infinite class X of finite non-abelian simple groups.     

The 2-adic affine building of type Ã2 and its finite projections

A certain “free” group U is constructed that is generated by three elements of order 3 which pairwise generate a Frobenius group of order 21 and it is shown that U operates regularly on the affine building of type $\text{A}\text{?}{}_2$ over the field of 2-adic numbers. As a result an infinite series of finite rank 3 geometries is obtained whose rank 2 residues are projective planes of order 2, and which possess a regular automorphism group isomorphic to SL₃(p) or SU₃(p) for some prime p.     

Finite embedding theorems for partial Latin squares,quasi-groups,and loops

In this paper we prove that a finite partial commutative (idempotent commutative) Latin square can be embedded in a finite commutative (idempotent commutative) Latin square. These results are then used to show that the loop varieties defined by any non-empty subset of the identities {x(xy) = y, (yx)x = y} and the quasi-group varieties defined by any non-empty subset of {x² = x, x(xy) = y, (yx)x = y}, except possibly {x(xy) = y, (yx)x = y}, have the strong finite embeddability property. It is then shown that the finitely presented algebras in these varities are residually finite, Hopfian, and have a solvable word problem.     

On the residual finiteness of descending HNN-extensions of groups

Let G be a group of finite generic rank, φ an injective endomorphism of the group G, and G(φ) the descending HNN-extension of G corresponding to the endomorphism φ. Let the index of the subgroup Gφ in G be finite and equal to n. It is proved that, if the group G is almost residually π-finite for some set π of primes coprime to n, then the group G(φ) is residually finite. This generalizes a series of known results, including the Wise-Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.     

Infimal convolution in Efimov-Ste?kin Banach spaces

Let (X,‖⋅‖) be a reflexive Banach space with Kadec-Klee norm. Let f:X→(−∞,+∞] be a function which is either Lipschitzian or is proper, bounded below, and lower semi-continuous. Then f is supported from below by residually many parabolas opening downward, that is, the infimal convolution of ‖⋅^‖2 and f is attained at residually many points of X.