252 research outputs found
Numerical calculation of three-point branched covers of the projective line
We exhibit a numerical method to compute three-point branched covers of the
complex projective line. We develop algorithms for working explicitly with
Fuchsian triangle groups and their finite index subgroups, and we use these
algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate
Computing Invariants of Simplicial Manifolds
This is a survey of known algorithms in algebraic topology with a focus on
finite simplicial complexes and, in particular, simplicial manifolds. Wherever
possible an elementary approach is chosen. This way the text may also serve as
a condensed but very basic introduction to the algebraic topology of simplicial
manifolds.
This text will appear as a chapter in the forthcoming book "Triangulated
Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure
Zeta functions of groups and rings
The representation growth of a T -group is polynomial. We study the rate of polynomial growth and the spectrum of possible growth, showing that any rational number ? can be realized as the rate of polynomial growth of a class 2 nilpotent T -group. This is in stark contrast to the related subject of subgroup growth of T -groups where it has been shown that the set of possible growth rates is discrete in Q. We derive a formula for almost all of the local representation zeta functions of a T2-group with centre of Hirsch length 2. A consequence of this formula shows that the representation zeta function of such a group is finitely uniform. In contrast, we explicitly derive the representation zeta function of a specific T2-group with centre of Hirsch length 3 whose representation zeta function is not finitely uniform. We give formulae, in terms of Igusa's local zeta function, for the subring, left-, right- and two-sided ideal zeta function of a 2-dimensional ring. We use these formulae to compute a number of examples. In particular, we compute the subring zeta function of the ring of ?integers in a quadratic number field
Generating derivative structures: Algorithm and applications
We present an algorithm for generating all derivative superstructures--for
arbitrary parent structures and for any number of atom types. This algorithm
enumerates superlattices and atomic configurations in a geometry-independent
way. The key concept is to use the quotient group associated with each
superlattice to determine all unique atomic configurations. The run time of the
algorithm scales linearly with the number of unique structures found. We show
several applications demonstrating how the algorithm can be used in materials
design problems. We predict an altogether new crystal structure in Cd-Pt and
Pd-Pt, and several new ground states in Pd-rich and Pt-rich binary systems
Deciding Isomorphy using Dehn fillings, the splitting case
We solve Dehn's isomorphism problem for virtually torsion-free relatively
hyperbolic groups with nilpotent parabolic subgroups.
We do so by reducing the isomorphism problem to three algorithmic problems in
the parabolic subgroups, namely the isomorphism problem, separation of torsion
(in their outer automorphism groups) by congruences, and the mixed Whitehead
problem, an automorphism group orbit problem. The first step of the reduction
is to compute canonical JSJ decompositions. Dehn fillings and the given
solutions of the algorithmic problems in the parabolic groups are then used to
decide if the graphs of groups have isomorphic vertex groups and, if so,
whether a global isomorphism can be assembled.
For the class of finitely generated nilpotent groups, we give solutions to
these algorithmic problems by using the arithmetic nature of these groups and
of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections.
The main changes to the previous version are a better treatment of the
algorithmic recognition and presentation of virtually cyclic subgroups and a
new proof of a rigidity criterion obtained by passing to a torsion-free
finite index subgroup. The previous proof relied on an incorrect result. To
appear in Inventiones Mathematica
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