Nicholas W. M. Touikan

Strong accessibility for finitely presented groups

A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost) finitely presented groups and show that it is finite, provided the group in question doesn’t contain any slender subgroups with infinite dihedral quotients and satisfies an ascending chain condition on certain chains of subgroups of edge groups. As a corollary, slender JSJ hierarchies of hyperbolic groups which are (virtually) without 2–torsion and finitely presented subgroups of SLn(Z) are both finite.

The Solvability Problem for Quadratic Equations over Free Groups is NP-Complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete

Detecting geometric splittings in finitely presented groups

We present an algorithm which given a presentation of a group

On the one-endedness of graphs of groups

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The equation w(x, y) = u over free groups: an algebraic approach

without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant

Multipass automata and group word problems

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On geodesic ray bundles in hyperbolic groups

, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if

Bulitko's Lemma for acylindrical splittings

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Deciding isomorphy using Dehn fillings, the splitting case

admits an essential free decomposition, the second is an algorithm which; if

Magnus pairs in, and free conjugacy separability of, limit groups

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