129,114 research outputs found
The Tits alternative for generalized triangle groups of type (3, 4, 2)
A generalized triangle group is a group that can be presented in the form G = h x, y | xp = yq = w(x, y)r = 1 i where p, q, r ? 2 and w(x, y) is a cyclically reduced word of length at least 2 in the free product Zp ? Zq = h x, y | xp = yq = 1i. Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p, q, r) is one of (2, 3, 2), (2, 4, 2), (2, 5, 2), (3, 3, 2), (3, 4, 2), or (3, 5, 2). Building on a result of Benyash-Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case (p, q, r) = (3, 4, 2)
A System Exhibiting Toroidal Order
A two dimensional system of discs upon which a triangle of spins are mounted
is shown to undergo a sequence of interesting phase transitions as the
temperature is lowered. We are mainly concerned with the `solid' phase in which
bond orientational order but not positional order is long ranged. As the
temperature is lowered in the `solid' phase, the first phase transition
involving the orientation or toroidal charge of the discs is into a `gauge
toroid' phase in which the product of a magnetic toroidal parameter and an
orientation variable (for the discs) orders but due to a local gauge symmetry
these variables themselves do not individually order. Finally, in the lowest
temperature phase the gauge symmetry is broken and toroidal order and
orientational order both develop. In the `gauge toroidal' phase time reversal
invariance is broken and in the lowest temperature phase inversion symmetry is
also broken. In none of these phases is there long range order in any Fourier
component of the average spin. A definition of the toroidal magnetic moment
of the th plaquette is proposed such that the magnetostatic
interaction between plaquettes and is proportional to .
Symmetry considerations are used to construct the magnetoelectric free energy
and thereby to deduce which coefficients of the linear magnetoelectric tensor
are allowed to be nonzero. In none of the phases does symmetry permit a
spontaneous polarization.Comment: 9 pages, 6 figure
New Beauville surfaces and finite simple groups
In this paper we construct new Beauville surfaces with group either
\PSL(2,p^e), or belonging to some other families of finite simple groups of
Lie type of low Lie rank, or an alternating group, or a symmetric group,
proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on
probabilistic group theoretical results of Liebeck and Shalev, on classical
results of Macbeath and on recent results of Marion.Comment: v4: 18 pages. Final version, to appear in Manuscripta Mat
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