Free-by -cyclic groups have been studied as algebraic counterparts of cusped hyperbolic mapping torus groups. Free-by-cyclic groups and cusped hyperbolic mapping torus groups share many algebraic properties. Nonetheless, free-by-cyclic groups are more complicated because not every free-by-cyclic group is realized as a cusped hyperbolic mapping torus group. In this talk, I explain some basic concepts and summarize some previous results related to free-by-cyclic groups. Also, I discuss some problems about free-by-cyclic groups.
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