Topology of <Emphasis Type="Italic">ϕ</Emphasis>-convex domains in calibrated manifolds

In [5], Harvey and Lawson showed that for any calibration ϕ there is an integer bound for the homotopy dimension of a strictly ϕ-convex domain and constructed a method to get these domains by using ϕ-free submanifolds. Here, we show how to get examples of ϕ-free submanifolds with different homotopy types for the quaternion calibration in ℍ ⁿ , associative calibration, and coassociative calibration in G ₂ manifolds. Hence we give examples of strictly ϕ-convex domains with different homotopy types allowed by Morse Theory.     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ. Partially supported by NSF grant DMS-0311142.     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

Metric properties of the tropical Abel–Jacobi map

Let Γ be a tropical curve (or metric graph), and fix a base point p∈Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel–Jacobi map Φ _p :Γ→J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and \varPhi_p^*(r){\varPhi}_{p}^{*}(\rho) for three different natural “metrics” ρ on J(Γ). One of these formulas implies that Φ _p is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μ _Zh on a metric graph Γ, defined by S. Zhang, measures lengths on Φ _p (Γ) with respect to the “sup-norm” on J(Γ).     

Groups with an element permuting with finitely many of its conjugates

We study the problem as to which is the cardinality of connected components of the graph Γ_α, defined as follows. Let G be a group and a an element of G. The vertex set V(Γ_α) of the graph is the conjugacy class of elements,Cl _G(a), and two vertices x and y of the graph Γ_α are bridged by an edge iff x=y. If the intersectionC _G(a)∩Cl _G(a) is finite, Γ_α is locally finite. We prove that connected components of the locally finite graph Γ_α are finite in some classes of groups. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996.     

On r-recognition by prime graph of Bp(3) where p is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G @ B_p(3){G\cong B_p(3)} or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then G @ B₃(3), C₃(3), D₄(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O₂(G) @ Aut(²B₂(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained.     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

Groups containing an element that permutes with a finite number of its conjugates

We study a group G containing an element g such that C_G(g)∩g^G is finite. The nonoriented graph Γ is defined as follows. The vertex set of Γ is the conjugacy class g^G. Vertices x and y of the graph G are bridged by an edge iff x≠y and xy=yx. Let Γ₀ be some connected component of G. On a condition that any two vertices of Γ₀ generate a nilpotent group, it is proved that a subgroup generated by the vertex set of Γ₀ is locally nilpotent. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 637–650, November–December, 1998.     

On the existence of spines for ${\mathbb{Q}}${\mathbb{Q}}-rank 1 groups

Let X = Γ \G/ K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has -rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D₀. We prove that D₀ is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ -conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ.     

The higher lower central series

The Baer invariants Γ _n (G) of a group are central extensions of the elementsγ _n (G) of the lower central series. We show that the inclusionsγ _n +1 ⊂γ _n can be lifted to functor morphisms Γ _n+1→Γ _n and a canonical Lie algebra, analogous to Lazard’s Lie algebra, can be constructed which is explicitly computable. This is applied in various ways.     

Quasirecognition by prime graph of F4(q) where q?=?2n?>?2

The prime graph of a finite group G is denoted by Γ(G). In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G has a unique nonabelian composition factor isomorphic to F ₄(q). We also show that if G is a finite group satisfying |G| = |F ₄(q)| and Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G @ F₄(q){G \cong F_4(q)}. As a consequence of our result we give a new proof for a conjecture of Shi and Bi for F ₄(q) where q = 2 ⁿ  > 2.     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998     

Logarithm laws for flows on homogeneous spaces

In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {A _t  | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f _t of G under which #{t∈ℕ | f _t x∈A _t } is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov. Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999     

Divergent trajectories and ℚ-rank

The author proves a conjecture of the author: IfG is a semisimple real algebraic defined over ℚ, Γ is an arithmetic subgroup (with respect to the given ℚ-structure) andA is a diagonalizable subgroup admitting a divergent trajectory inG/Γ, then dimA≤ rank_ℚ G.     

Distribution of lattice orbits on homogeneous varieties

Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H\ G. For an increasing family of finite subsets {Γ _T : T > 0}, a dense orbit υ· Γ, υ∈V and compactly supported function φ on V, we consider the sums . Understanding the asymptotic behavior of S _φ,υ (T) is a delicate problem which has only been considered for certain very special choices of H,G and {Γ _T }. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have where G _T = {g ∈G: ||g|| < T} and Γ _T = G _T ∩ Γ. We also show that the asymptotics of S _{φ, υ} (T) is governed by where ν is an explicit limiting density depending on the choice of υ and || · ||. Submitted: March 2005 Revision: April 2006 Accepted: June 2006     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

A Family of Non-quasiprimitive Graphs Constructed From PSU3（q） Where q Is Odd

Let Г be a simple connected graph and let G be a group of automorphisms of Г. Г is said to be （G, 2）-arc transitive if G is transitive on the 2-arcs of Г. It has been shown that there exists a family of non-quasiprimitive （PSU3（q）, 2）-arc transitive graphs where q = 2^3m with m an odd integer. In this paper we investigate the case where q is an odd prime power.