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Register a new userPractical computation with linear groups over infinite domains
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We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algorithms have been used to perform extensive computer experiments.
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For an infinite field $F$, we study the cokernel of the map of homology groups $H_{n+1}(mathrm{GL}_{n-1}(F),mathbb{k}) to H_{n+1}(mathrm{GL}_{n}(F),mathbb{k})$, where $mathbb{k}$ is a field such that $(n-2)!in mathbb{k}^times$, and the kernel of the natural map $H_{n}big(mathrm{GL}_{n-1}(F),mathbb{Z}big[frac{1}{(n-2)!} big] big) to H_{n}big(mathrm{GL}_{n}(F),mathbb{Z}big[frac{1}{(n-2)!} big]big)$. We give conjectural estimates of these cokernel and kernel and prove our conjectures for $nleq 4$.
Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnors sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups of direct limit groups; (5) Constructions of Lie group structures on ascending unions of Lie groups.
We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n) in {(7 + 5k,5), (8 + 5k,5)} where k is greater than or equal to 0. This together with previously known results yields a complete classification of the finite F(r,n), a problem that has its origins in a question by J H Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.
Let $Gamma$ be a torsion-free hyperbolic group. We study $Gamma$--limit groups which, unlike the fundamental case in which $Gamma$ is free, may not be finitely presentable or geometrically tractable. We define model $Gamma$--limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary $Gamma$--limit group $L$, we canonically construct a strict resolution of a model $Gamma$--limit group, which encodes all homomorphisms $Lto Gamma$ that factor through the given resolution. We propose this as the correct framework in which to study $Gamma$--limit groups algorithmically. We enumerate all $Gamma$--limit groups in this framework.
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H leq mathrm{SL}(n, mathbb{Z})$ for $n geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $mathrm{SL}(n, mathbb{Q})$ for $n > 2$.
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