Larsen Louder

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.

Krull dimension for limit groups

We show that varieties defined over free groups have finite Krull dimension, answering a question of Z Sela.

Decision problems, complexity, traces, and representations

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on infinitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinct pair of conjugacy classes.

Scott complexity and adjoining roots to finitely generated groups

We give several generalizations of the well-known fact that a commutator in a free group is not a proper power.

Zariski closures and subgroup separability

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group.

Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups

We use hyperbolic towers to answer some model-theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type p0 but that there is a finitely generated model which omits p(2)0. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.

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

There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable.

Three techniques for obtaining algebraic circle packings

The main purpose of this article is to demonstrate three techniques for proving algebraicity statements about circle packings. We give proofs of three related theorems: (1) that every finite simple planar graph is the contact graph of a circle packing on the Riemann sphere, equivalently in the complex plane, all of whose tangency points, centers, and radii are algebraic, (2) that every flat conformal torus which admits a circle packing whose contact graph triangulates the torus has algebraic modulus, and (3) that if R is a compact Riemann surface of genus at least 2, having constant curvature -1, which admits a circle packing whose contact graph triangulates R, then R is isomorphic to the quotient of the hyperbolic plane by a subgroup of PSL_2(real algebraic numbers). The statement (1) is original, while (2) and (3) have been previously proved in the Ph.D. thesis of McCaughan. Our first proof technique is to apply Tarskis Theorem, a result from model theory, which says that if an elementary statement in the theory of real-closed fields is true over one real-closed field, then it is true over any real closed field. This technique works to prove (1) and (2). Our second proof technique is via an algebraicity result of Thurston on finite co-volume discrete subgroups of the orientation-preserving-isometry group of hyperbolic 3-space. This technique works to prove (1). Our first and second techniques had not previously been applied in this area. Our third and final technique is via a lemma in real algebraic geometry, and was previously used by McCaughan to prove (2) and (3). We show that in fact it may be used to prove (1) as well.