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We consider linear slices of the space of Kleinian once-punctured torus groups; a linear slice is obtained by fixing the value of the trace of one of the generators. The linear slice for trace 2 is called the Maskit slice. We will show that if traces converge ‘horocyclically’ to 2 then associated linear slices converge to the Maskit slice, whereas if the traces converge ‘tangentially’ to 2 the linear slices converge to a proper subset of the Maskit slice. This result will be also rephrased in terms of complex Fenchel–Nielsen coordinates. In addition, we will show that there is a linear slice which is not locally connected.

Linear slices close to a Maskit slice