Subset currents on free groups

We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure on a "branching" analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$.Comment: updated version; to appear in Geometriae Dedicat

Stabilizers of R\mathbb R-trees with free isometric actions of FNF_N

We prove that if $T$ is an $\mathbb R$-tree with a minimal free isometric action of $F_N$, then the $Out(F_N)$-stabilizer of the projective class $[T]$ is virtually cyclic. For the special case where $T=T_+(\phi)$ is the forward limit tree of an atoroidal iwip element $\phi\in Out(F_N)$ this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip (not necessarily atoroidal): we prove that every such subgroup $G\le Out(F_N)$ is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of $Out(F_N)$ is due to Bestvina, Feighn and Handel.Comment: corrected the proof of Proposition 4.1, plus several minor fixes and updates; to appear in Journal of Group Theor

Intersection form, laminations and currents on free groups

Let $F_N$ be a free group of rank $N\ge 2$, let $\mu$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $$ is equal to zero if and only if the support of $\mu$ is contained in the dual algebraic lamination $L^2(T)$ of $T$. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \emph{filling} element in $F_N$ and prove that filling elements are "nearly generic" in $F_N$. We also apply our results to the notion of \emph{bounded translation equivalence} in free groups.Comment: revised version, to appear in GAF

Conjugacy in Baumslag's group, generic case complexity, and division in power circuits

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS(1,2) and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is TC^0-complete. To the best of our knowledge BS(1,2) is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to show that conjugacy in G(1,2) can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G(1,2) can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G(1,2) by reducing the division problem in power circuits to the conjugacy problem in G(1,2). The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 = {\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the base group H if and only if A \neq H \neq

A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System

This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J and a new non-local, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R (obtained by composing K and the inverse of J.) In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi-Hamiltonian structures for other families of discrete, integrable equations.Comment: 33 page

Conformal dimension and random groups

We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where $l$ is the relator length, going to infinity. (a) 1 + 1/C < \Cdim(\bdry G) < C l / \log(l), for the few relator model, and (b) 1 + l / (C\log(l)) < \Cdim(\bdry G) < C l, for the density model, at densities $d < 1/16$. In particular, for the density model at densities $d < 1/16$, as the relator length $l$ goes to infinity, the random groups will pass through infinitely many different quasi-isometry classes.Comment: 32 pages, 4 figures. v2: Final version. Main result improved to density < 1/16. Many minor improvements. To appear in GAF

Twisted geometries: A geometric parametrisation of SU(2) phase space

A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this paper we show how to parametrize this phase space in terms of quantities describing the intrinsic and extrinsic geometry of the triangulation dual to the graph. These are defined by the assignment to each triangle of its area, the two unit normals as seen from the two polyhedra sharing it, and an additional angle related to the extrinsic curvature. These quantities do not define a Regge geometry, since they include extrinsic data, but a looser notion of discrete geometry which is twisted in the sense that it is locally well-defined, but the local patches lack a consistent gluing among each other. We give the Poisson brackets among the new variables, and exhibit a symplectomorphism which maps them into the Poisson brackets of loop gravity. The new parametrization has the advantage of a simple description of the gauge-invariant reduced phase space, which is given by a product of phase spaces associated to edges and vertices, and it also provides an abelianisation of the SU(2) connection. The results are relevant for the construction of coherent states, and as a byproduct, contribute to clarify the connection between loop gravity and its subset corresponding to Regge geometries.Comment: 28 pages. v2 and v3 minor change

Length functions on currents and applications to dynamics and counting

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-Anosov homeomorphisms of closed hyperbolic surfaces act on the space of projective geodesic currents with uniform north-south dynamics.Comment: 35pp, 2 figures, comments welcome! Second version: minor corrections. To appear as a chapter in the forthcoming book "In the tradition of Thurston" edited by V. Alberge, K. Ohshika and A. Papadopoulo

Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra U(\g) of a semisimple Lie algebra \g. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.Comment: 11 pages, references adde

Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat