Armando Martino

Metric properties of outer space

We define metrics on Culler-Vogtmann space, which are an analogue of the Thurston metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer Space, quasi-geodesic for the symmetric metric.

Orbit decidability and the conjugacy problem for some extensions of groups

Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, we prove that G has solvable conjugacy problem if and only if the corresponding action subgroup A 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2) is given.

Conjugacy in normal subgroups of hyperbolic groups

Abstract. Let N be a finitely generated normal subgroup of a Gromov hyperbolic group . We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of . We show that the hyperbolic group from F. Haglunds and D. Wises version of Ripss construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index.

The isometry group of outer space

Abstract We prove analogues of Royden’s Theorem for the Lipschitz metrics of Outer Space, namely that Isom ( CV n ) = Out ( F n ) .

Fixed Subgroups are Compressed in Free Groups

Abstract In this paper, we prove that the fixed subgroup of an arbitrary family of endomorphisms ψ i , i ∈ I, of a finitely generated free group F, is F-super-compressed. In particular, r(∩ i∈I Fix ψ i ) ≤ r(M) for every subgroup M ≤ F containing ∩ i∈I Fix ψ i . This provides new evidence towards the inertia conjecture for fixed subgroups of free groups. As a corollary, we show that, in the free group of rank n, every strictly ascending chain of fixed subgroups has length at most 2n. This answers a question of Levitt.

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

Commensurations and metric properties of Houghton’s groups

We describe the automorphism groups and the abstract commensurators of Houghtons groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.

The Automorphism Group of a Free-by-Cyclic Group in Rank 2

Let φ be an automorphism of a free group Fn of rank n, and let Mφ = Fn ⋊φ ℤ be the corresponding mapping torus of φ. We study the group Out(Mφ) under certain technical conditions on φ. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of φ in GL2(ℤ). As an application, we solve the isomorphism problem for the family of F2-by-ℤ groups, in terms of the two defining automorphisms.

Conjugacy in Houghton's groups

Let n ∈ N. Houghton’s group Hn is the group of permutations of {1, . . . , n} × N, that eventually act as a translation in each copy of N. We prove the solvability of the conjugacy problem and conjugator search problem for Hn, n ≥ 2.

Examples of retracts in free groups that are not the fixed subgroup of any automorphism

A knee joint prosthesis for permanent anchoring in the bone tissue of a knee joint including a femur and a tibia is disclosed including a femur portion having an articulation element with a convex surface, a fixation rail for anchoring the articulation element to the femur, and an anchoring element for anchoring the fixation rail to the femur, and the tibia portion including a movable tibia plateau having a concave surface for cooperation with the convex surface of the femur articulation element, a fixed tibia plateau for anchoring the movable tibia plateau to the tibia, and an anchoring element for anchoring the movable tibia plateau to the tibia, the anchoring element including a pair of cylindrical rods imbedded in the tibia transversely to the longitudinal direction of the tibia and including at least one recess, with the fixed tibia plateau including a locking base which can be received by the recess in the cylindrical rods to anchor the movable tibia plateau to the anchoring element.