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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.
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
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 m
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
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 alway
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 g