The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group.Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. On a related note, we prove that any non-trivial group which can be approximated by finite groups has a non-trivial quotient that can be approximated by finite special linear groups.Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that a the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well.All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal. In Section 4 we also use results of Liebeck and Shalev on bounded generation in finite simple groups.

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The decarbonization of the transport sector, and thus of road-based transport logistics, through electrification, is essential to achieve European climate targets. Battery electric trucks offer the greatest well-to-wheel potential for CO2 saving. At the same time, however, they are subject to restrictions due to charging events because of their limited range compared to conventional trucks. These restrictions can be kept to a minimum through optimal charging stop strategies. In this paper, we quantify these restrictions and show the potential of optimal strategies. The modeling of an optimal charging stop strategy is described mathematically as an optimization problem and solved by a genetic algorithm. The results show that in the case of long-distance transport using trucks with battery capacities lower than 750 kWh, a time loss is to be expected. However, this can be kept below 20 min for most battery capacities by optimal charging stops and sufficient charging infrastructure.