Abelian coverings of finite general linear groups and an application to their non-commuting graphs

In this paper we introduce and study a family A_n(q)\mathcal{A}_{n}(q) of abelian subgroups of GL_n(q){\rm GL}_{n}(q) covering every element of GL_n(q){\rm GL}_{n}(q). We show that A_n(q)\mathcal{A}_{n}(q) contains all the centralizers of cyclic matrices and equality holds if q>n. For q>2, we obtain an infinite product expression for a probabilistic generating function for |A_n(q)||\mathcal{A}_{n}(q)|. This leads to upper and lower bounds which show in particular that

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

This paper continues the investigation of the groups RF(G)\mathcal{RF}(G) first introduced in the forthcoming book of Chiswell and Müller “A Class of Groups Universal for Free ℝ-Tree Actions” and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009). We establish a criterion for a family {H_s}\{\mathcal{H}_{\sigma}\} of hyperbolic subgroups H_s ￡ RF(G)\mathcal{H}_{\sigma}\leq\mathcal{RF}(G) to generate a hyperbolic subgroup isomorphic to the free product of the H_s\mathcal{H}_{\sigma} (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in RF(G)\mathcal{RF}(G) for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in RF(G)\mathcal{RF}(G) generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).     

Geometric invariant theory and the generalized eigenvalue problem

Let G be a connected reductive subgroup of a complex connected reductive group [^(G)]\hat{G}. Fix maximal tori and Borel subgroups of G and [^(G)]{\hat{G}}. Consider the cone LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) generated by the pairs (n,[^(n)])(\nu,{\hat{\nu}}) of dominant characters such that V_n^*V_{\nu}^{*} is a submodule of V_[^(n)]V_{{\hat{\nu}}} (with usual notation). Here we give a minimal set of inequalities describing LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) as a part of the dominant chamber. In other words, we describe the facets of LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) which intersect the interior of the dominant chamber. We also describe smaller faces. Finally, we are interested in some classical redundant inequalities.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Pan-H-Linked Graphs

Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n ₀(H) isolated vertices and n ₁(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165–182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H{\mathcal{H}} and d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}}, then G contains a spanning H-subdivision with the same ground set as H{\mathcal{H}} . As a corollary to this result, the authors were able to obtain Dirac’s famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3 with d(G) 3 \fracn2{\delta(G)\ge\frac{n}{2}} , then G is hamiltonian. Bondy (J. Comb. Theory Ser. B 11:80–84, 1971) extended Dirac’s theorem by showing that if G satisfied the condition d(G) 3 \fracn2{\delta(G) \ge \frac{n}{2}} then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165–182, 2007) in a similar manner. An H-subdivision H{\mathcal{H}} in G is 1-extendible if there exists an H-subdivision H^*{\mathcal{H}^{*}} with the same ground set as H{\mathcal{H}} and |H^*| = |H| + 1{|\mathcal{H}^{*}| = |\mathcal{H}| + 1} . If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}} , then G is pan-H-linked. This result is sharp.     

On Representations of Integers in Thin Subgroups of {{\rm SL}_2({\mathbb {Z}})}

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ ? \mathbb Z²  \  {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv₀g, w₀?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on \mathbb R²{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set \mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S   (mod  q)   for all   q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings, |\mathfrak E(N)| << N^1-e₀{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞.     

Convergence and Stability of Locally ℝ N -Invariant Solutions of Ricci Flow

Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally $\mathcal{G}Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G\mathcal{G} -invariant solutions on bundles G^N\hookrightarrowM \oversetp? Bⁿ\mathcal{G}^{N}\hookrightarrow\mathcal{M}\,\overset{\pi }{\mathcal{\longrightarrow}}\,\mathcal{B}^{n} , with G\mathcal{G} a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain ℝ ^N -invariant model solutions. In case N+n=3, our results are relevant to work of Lott classifying the asymptotic behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures and diameters are respectively O(t^-1)\mathcal{O}(t^{-1}) and O(t^1/2)\mathcal{O}(t^{1/2}) as t→∞.     

The noncommutative Choquet boundary II: Hyperrigidity

A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space A ⊆ B(H) and every sequence of unital completely positive linear maps ϕ ₁, ϕ ₂,... from B(H) to itself,

On some inequalities for Doob decompositions in Banach function spaces

Let ${\Phi : \mathbb{R} \to [0, \infty)}Let F: \mathbbR ? [0, ￥){\Phi : \mathbb{R} \to [0, \infty)} be a Young function and let f = (f_n)_n ? \mathbbZ₊{f = (f_n)_n\in\mathbb{Z}_{+}} be a martingale such that F(f_n) ? L₁{\Phi(f_n) \in L_1} for all n ? \mathbbZ₊{n \in \mathbb{Z}_{+}} . Then the process F(f) = (F(f_n))_n ? \mathbbZ₊{\Phi(f) = (\Phi(f_n))_n\in\mathbb{Z}_{+}} can be uniquely decomposed as F(f_n)=g_n+h_n{\Phi(f_n)=g_n+h_n} , where g=(g_n)_n ? \mathbbZ₊{g=(g_n)_n\in\mathbb{Z}_{+}} is a martingale and h=(h_n)_n ? \mathbbZ₊{h=(h_n)_n\in\mathbb{Z}_{+}} is a predictable nondecreasing process such that h ₀ = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality ||h_￥||_X ￡ C ||F(Mf) ||_X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Mf)} \|_X} is valid, and those X such that the inequality ||h_￥||_X ￡ C ||F(Sf) ||_X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Sf)} \|_X} is valid, where Mf and Sf denote the maximal function and the square function of f, respectively.     

The Cycle Discrepancy of Three-Regular Graphs

Let G = (V, E) be an undirected graph and C(G){{\mathcal C}(G)} denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as

Idempotents in a complex group algebra, projective modules, and the von Neumann algebra

The complex group algebra \Bbb CG{\Bbb C}G of a countable group G can be imbedded in the von Neumann algebra NG of G. If G is torsion-free, and if P is a finitely generated projective module over \Bbb CG{\Bbb C}G it is proved that the central-valued trace of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P, i.e. of an idempotent \Bbb CG{\Bbb C}G-matrix A defining P is equal to the canonical trace k(P)\kappa (P) times identity I. It follows that k(P)\kappa (P) characterizes the isomorphism type of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P.¶If k(P)\kappa (P) is an integer, e.g., if the weak Bass conjecture holds for G then NG?_{\Bbb C G}PNG\otimes _{{\Bbb C} G}P is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.     

On quasiconvexity and relatively hyperbolic structures on groups

Let G be a group which is hyperbolic relative to a collection of subgroups H₁{\mathcal{H}_1}, and it is also hyperbolic relative to a collection of subgroups H₂{\mathcal{H}_2}. Suppose that H₁ ì H₂{\mathcal{H}_1 \subset \mathcal{H}_2}. We characterize when a relative quasiconvex subgroup of (G, H₂){(G, \mathcal H_2)} is still relatively quasiconvex in (G, H₁){(G, \mathcal H_1)}. We also show that relative quasiconvexity is preserved when passing from (G, H₁){(G, \mathcal H_1)} to (G, H₂){(G, \mathcal H_2)}. Applications are discussed.     

Uniformization of {\mathcal {G}}-bundles

We show some of the conjectures of Pappas and Rapoport concerning the moduli stack Bun_G{{\rm Bun}_\mathcal {G}} of G{\mathcal {G}}-torsors on a curve C, where G{\mathcal {G}} is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of Bun_G{{\rm Bun}_\mathcal {G}} in case G{\mathcal {G}} is simply connected.     

On the singularity of the irreducible components of a Springer fiber in {\mathfrak{s}\mathfrak{l}_n}

Let ${\mathcal{B}_u}Let B_u{\mathcal{B}_u} be the Springer fiber over a nilpotent endomorphism u ? End(\mathbbCⁿ){u\in {\rm End}(\mathbb{C}^n)}. Let J (u) be the Jordan form of u regarded as a partition of n. The irreducible components of B_u{\mathcal{B}_u} are all of the same dimension. They are labelled by Young tableaux of shape J (u). We study the question of the singularity of the components of B_u{\mathcal{B}_u} and show that all the components of B_u{\mathcal{B}_u} are nonsingular if and only if J(u) ? {(l,1,1,?), (l₁,l₂), (l₁,l₂,1), (2,2,2)}{J(u)\in\{(\lambda,1,1,\ldots), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}}.     

Maximum Hitting of a Set by Compressed Intersecting Families

For a family A{\mathcal{A}} and a set Z, denote {A ? A \colon A ?Z 1 ?}{\{A \in \mathcal{A} \colon A \cap Z \neq \emptyset\}} by A(Z){\mathcal{A}(Z)}. For positive integers n and r, let S_n,r{\mathcal{S}_{n,r}} be the trivial compressed intersecting family {A ? (c[n]r ) \colon 1 ? A}{\{A \in \big(\begin{subarray}{c}[n]\\r \end{subarray}\big) \colon 1 \in A\}}, where [n] : = {1, ?, n}{[n] := \{1, \ldots, n\}} and (c[n]r ) : = {A ì [n] \colon |A| = r}{\big(\begin{subarray}{c}[n]\\r \end{subarray}\big) := \{A \subset [n] \colon |A| = r\}}. The following problem is considered: For r ≤ n/2, which sets Z í [n]{Z \subseteq [n]} have the property that |A(Z)| ￡ |S_n,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} for any compressed intersecting family A ì (c[n]r ){\mathcal{A}\subset \big(\begin{subarray}{c}[n]\\r \end{subarray}\big)}? (The answer for the case 1 ? Z{1 \in Z} is given by the Erdős–Ko–Rado Theorem.) We give a complete answer for the case |Z| ≥ r and a partial answer for the much harder case |Z| < r. This paper is motivated by the observation that certain interesting results in extremal set theory can be proved by answering the question above for particular sets Z. Using our result for the special case when Z is the r-segment {2, ?, r+1}{\{2, \ldots, r+1\}}, we obtain new short proofs of two well-known Hilton–Milner theorems. At the other extreme end, by establishing that |A(Z)| ￡ |S_n,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} when Z is a final segment, we provide a new short proof of a Holroyd–Talbot extension of the Erdős-Ko-Rado Theorem.     

On {\phi}-contractibility of the Lebesgue–Fourier algebra of a locally compact group

For a locally compact group G, we present some characterizations for f{\phi}-contractibility of the Lebesgue–Fourier algebra LA(G){\mathcal{L}A(G)} endowed with convolution or pointwise product.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

Connected Domination Number of a Graph and its Complement

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γ _c (G) is the minimum size of such a set. Let d^*(G)=min{d(G),d([`(G)])}{\delta^*(G)={\rm min}\{\delta(G),\delta({\overline{G}})\}} , where [`(G)]{{\overline{G}}} is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and [`(G)]{{\overline{G}}} are both connected, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ d^*(G)+4-(g_c(G)-3)(g_c([`(G)])-3){{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+4-({\gamma_c}(G)-3)({\gamma_c}({\overline{G}})-3)} . As a corollary, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ \frac3n4{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \frac{3n}{4}} when δ*(G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that g_c(G)+g_c([`(G)]) ￡ d<sup>*</sup>(G)+2{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+2} when g<sub>c</sub>(G),g<sub>c</sub>([`(G)]) 3 4{{\gamma_c}(G),{\gamma_c}({\overline{G}})\ge 4} . This bound is sharp when δ*(G) = 6, and equality can only hold when δ*(G) = 6. Finally, we prove that g_c(G)g_c([`(G)]) ￡ 2n-4{{\gamma_c}(G){\gamma_c}({\overline{G}})\le 2n-4} when n ≥ 7, with equality only for paths and cycles.     

Delsarte Set Graphs with Small c 2

Let Γ be a Delsarte set graph with an intersection number c ₂ (i.e., a distance-regular graph with a set ${\mathcal{C}}Let Γ be a Delsarte set graph with an intersection number c ₂ (i.e., a distance-regular graph with a set C{\mathcal{C}} of Delsarte cliques such that each edge lies in a positive constant number n_C{n_{\mathcal{C}}} of Delsarte cliques in C{\mathcal{C}}). We showed in Bang et al. (J Combin 28:501–506, 2007) that if ψ ₁ > 1 then c ₂ ≥ 2 ψ ₁ where y₁:=|G₁(x)?C |{\psi_1:=|\Gamma_1(x)\cap C |} for x ? V(G){x\in V(\Gamma)} and C a Delsarte clique satisfying d(x, C) = 1. In this paper, we classify Γ with the case c ₂ = 2ψ ₁ > 2. As a consequence of this result, we show that if c ₂ ≤ 5 and ψ ₁ > 1 then Γ is either a Johnson graph or a folded Johnson graph [`(J)](4s,2s){\overline{J}(4s,2s)} with s ≥ 3.     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.