Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

L-spaces, taut foliations, and graph manifolds

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

Quasi-isometries of pairs: Surfaces in graph manifolds

We show there exists a closed graph manifold [Formula: see text] and infinitely many non-separable, horizontal surfaces [Formula: see text] such that there does not exist a quasi-isometry [Formula: see text] taking [Formula: see text] to [Formula: see text] within a finite Hausdorff distance when [Formula: see text].