Hybrid journal (It can contain Open Access articles) ISSN (Print) 0024-6093 - ISSN (Online) 1469-2120 Published by Oxford University Press[406 journals]

Authors:Gillespie J. Pages: 895 - 922 Abstract: We discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories. PubDate: Tue, 13 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw051 Issue No:Vol. 48, No. 6 (2016)

Authors:Oliver B. Pages: 923 - 934 Abstract: We prove that if ${\mathcal {E}}\trianglelefteq {\mathcal {F}}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S({\mathcal {E}})\le T$, and either ${\rm Aut}_{{\mathcal {F}}}(T)/{\rm Aut}_{\mathcal {E}}(T)$ or ${\rm Out}({\mathcal {E}})$ is $p$-solvable, then ${\mathcal {F}}$ can be ‘reduced’ to ${\mathcal {E}}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever ${\mathcal {E}}$ is simple and ‘tamely realized’ by a known simple group $K$. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems. PubDate: Thu, 06 Oct 2016 00:00:00 GMT DOI: 10.1112/blms/bdw052 Issue No:Vol. 48, No. 6 (2016)

Authors:Keller M; Lenz D, Münch F, et al. Pages: 935 - 944 Abstract: We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan-type behavior on manifolds. Moreover, this also gives that short-time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded. PubDate: Thu, 03 Nov 2016 00:00:00 GMT DOI: 10.1112/blms/bdw054 Issue No:Vol. 48, No. 6 (2016)

Authors:Akita T. Pages: 945 - 956 Abstract: Given an odd prime number $p$ and a Coxeter group $W$ such that the order of the product $st$ is prime to $p$ for all Coxeter generators $s,t$ of $W$, we prove that the $p$-local homology groups $H_k(W,\mathbb {Z}_{{(p)}})$ vanish for $1\leq k\leq 2(p-2)$. This generalizes a known vanishing result for symmetric groups due to Minoru Nakaoka. PubDate: Wed, 19 Oct 2016 00:00:00 GMT DOI: 10.1112/blms/bdw063 Issue No:Vol. 48, No. 6 (2016)

Authors:Baba S; Granath H. Pages: 957 - 967 Abstract: The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic $(2,2,2,3)$ quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura curve and relate the ring generators to explicit Heun functions for the quadrilateral group. We also show that the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. These results are used to completely describe the exceptional sets of the Heun functions, and we show how to find examples like \[ Hl\left(\frac{27}{2},\frac{7}{36}; \frac{1}{12},\frac{7}{12},\frac{2}{3},\frac{1}{2}; -\frac{96}{25}\right)=\frac{2^{1/2}5^{2/3}}{3^{4/3}}. \] PubDate: Fri, 16 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw055 Issue No:Vol. 48, No. 6 (2016)

Authors:Losert V. Pages: 968 - 976 Abstract: For a discrete group $G$ with Fourier algebra $A(G)$, we study the topological centre $\mathfrak {Z}_t$ of the bidual $A(G)''$. If $G$ is amenable, then $\mathfrak {Z}_t=A(G)$. But if $G$ contains a free group $F_r$ ($r\ge 2$), we show that $\mathfrak {Z}_t$ is strictly larger than $A(G)$. Furthermore, it is shown that the subalgebra $A_\# (G)$ of radial functions in $A(G)$ is Arens regular. PubDate: Fri, 16 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw053 Issue No:Vol. 48, No. 6 (2016)

Authors:Martínez Torres D. Pages: 977 - 984 Abstract: Semisimple (co)adjoint orbits through real hyperbolic elements are well known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both the Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and the symplectic geometry of real hyperbolic (co)adjoint orbits. PubDate: Fri, 16 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw058 Issue No:Vol. 48, No. 6 (2016)

Authors:Krashen D. Pages: 985 - 1000 Abstract: We use constructions of versal cohomology classes based on a new notion of ‘presentable functors’, to describe a relationship between the problems of bounding symbol length in cohomology and of finding the minimal degree of a splitting field. The constructions involved are then also used to describe generic splitting varieties for degree 2 cohomology with coefficients in a commutative algebraic group of multiplicative type. PubDate: Thu, 03 Nov 2016 00:00:00 GMT DOI: 10.1112/blms/bdw060 Issue No:Vol. 48, No. 6 (2016)

Authors:Kropholler R; Wilkes G. Pages: 1001 - 1007 Abstract: In this paper, we prove that right-angled Artin groups (RAAGs) are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that right-angled Coxeter groups (RACGs) are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore, we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion. PubDate: Fri, 16 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw056 Issue No:Vol. 48, No. 6 (2016)

Authors:Amoroso F; Masser D. Pages: 1008 - 1012 Abstract: We prove close to sharp lower bounds for the height of an algebraic number in a Galois extension of $\mathbb {Q}$. PubDate: Fri, 16 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw057 Issue No:Vol. 48, No. 6 (2016)

Authors:Conder M; Širáň J. Pages: 1013 - 1017 Abstract: We prove that for every integer $d\ge 3$ and every group $U$ of units mod $d,$ there exists an orientably-regular map of valency $d$ with exponent group $U$. PubDate: Fri, 16 Sep 2016 00:00:00 GMT DOI: 10.1112/blms/bdw059 Issue No:Vol. 48, No. 6 (2016)

Authors:Walker A. Pages: 1018 - 1028 Abstract: The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group $(\mathbb {Z}/ q\mathbb {Z})^{\times }$ and prove a similar result. For all suitably large primes $q$ we define $P_\eta $ to be the set of primes less than $\eta q$, viewed naturally as a subset of $(\mathbb {Z}/ q\mathbb {Z})^{\times }$. Considering the $k$-fold product set $P_\eta ^{(k)}=\{p_1p_2\cdots p_k:p_i\in P_\eta \}$, we show that, for $\eta \gg q^{-{1}/{4}+\epsilon },$ there exists a constant $k$ depending only on $\epsilon $ such that $P_\eta ^{(k)}=(\mathbb {Z}/ q\mathbb {Z})^{\times }$. Erdös conjectured that, for $\eta = 1,$ the value $k=2$ should suffice: although we have not been able to prove this conjecture, we do establish that $P_1 ^{(2)}$ has density at least $\frac {1}{64}(1+o(1))$. We also formulate a similar theorem in almost-primes, improving on existing results. PubDate: Tue, 25 Oct 2016 00:00:00 GMT DOI: 10.1112/blms/bdw062 Issue No:Vol. 48, No. 6 (2016)

Authors:Hindes W. Pages: 1029 - 1036 Abstract: Given a polynomial $\phi $ over a global function field $K/\mathbb {F}_q(t)$ and a wandering base point $b\in K$, we give a geometric condition on $\phi ,$ ensuring the existence of primitive prime divisors for almost all points in the orbit ${\mathcal O}_\phi (b):=\{\phi ^n(b)\}_{n\geq 0}$. As an application, we prove that the Galois groups (over $K$) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of ${\mathcal O}_\phi (b)$. PubDate: Wed, 19 Oct 2016 00:00:00 GMT DOI: 10.1112/blms/bdw061 Issue No:Vol. 48, No. 6 (2016)

Authors:Macpherson D; Tent K. Pages: 1037 - 1049 Abstract: We consider profinite groups as 2-sorted first-order structures, with a group sort, and a second sort that acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of all open subgroups, then the first-order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index that is a direct product of finitely many compact $p$-adic analytic groups, for distinct primes $p$. In fact, the condition NIP can here be weakened to NTP${}_2$. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup. PubDate: Wed, 19 Oct 2016 00:00:00 GMT DOI: 10.1112/blms/bdw064 Issue No:Vol. 48, No. 6 (2016)