Groups of cohomological codimension one

We show that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> is a commensurated subgroup of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> such that both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> are of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mi>F</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">vcd</mml:mo> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo form="prefix">vcd</mml:mo> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is the fundamental group of a graph of groups in which all vertex and edge groups are commensurable to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> . We also investigate commensurated subgroups of one-relator groups and duality groups.