Publications and preprints

We study a skew product transformation associated to an irrational rotation of the circle [0,1]/~. This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of {0,1/2} in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is an infinite clique of 2-filling rays based at that end. These 2-filling rays are relevant to Bavard--Walker's loop graphs.

The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. See my notes here for a brief introduction to related problems in topology and group theory. We introduce new methods from stable commutator length to study this type of problems about nontriviality of one-relator quotients. Roughly, we show that surfaces in certain HNN extensions bounding a given word have complexity no less than the complexity of its boundary. A consequence of this is a Freiheitssatz theorem for HNN extensions, which in particular implies and gives a new proof of Klyachko's theorem that confirms the Kervaire conjecture for torsion-free groups. As another application, we also generalize the following theorem of Klyachko-Lurye to HNN extensions: For any group G and the quotient Q of G*Z by any proper power w^m with w in G*Z projecting to 1 in Z, the natural map G->Q is injective.

We give a short proof of Breuillard's height gap theorem using almost laws, which are word maps whose image lies close to the identity. The height gap theorem can be thought of as a non-abelian analog of Lehmer's Mahler measure problem and has nice applications.

Let S be a surface and let Mod(S,K) be the mapping class group of S permuting a Cantor subset K of S. We prove two structure theorems for normal subgroups of Mod(S,K).
(Purity:) if S has finite type, every normal subgroup of Mod(S,K) either contains the kernel of the forgetful map to the mapping class group of S, or it is `pure', i.e. it fixes the Cantor set pointwise.
(Inertia:) for any n element subset Q of the Cantor set, there is a forgetful map from the pure subgroup PMod(S,K) of Mod(S,K) to the mapping class group of (S,Q) fixing Q pointwise. If N is a normal subgroup of Mod(S,K) contained in PMod(S,K), its image N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N_Q arise this way.
Several applications and numerous examples are also given.

The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite abelian groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples.

We show groups with spectral gaps for scl of integral chains are preserved under taking finite graph products. In particular, RAAGs and RACGs all have spectral gaps for integral chains. Surprisingly the gap cannot be uniform over all RAAGs although there is a uniform spectral gap 1/2 for elements in RAAGs.
In a second part of this paper we show certain integral chains in RAAGs can be computed by linear programming and relate them to the fractional stability number of graphs. In particular this implies that computing scl of elements and chains in RAAGs is NP hard as well as other consequences.
All results are proved in the general setting of graph products. In particular all above results hold verbatim for RACGs.

We construct the so-called 2-filling rays on surfaces of infinite type, answering a question of Bavard-Walker. These rays are related to the understanding of the Gromov boundary of the loop graph of the surface in terms of geodesic laminations. We give a hands-on contruction as well as one using train tracks and laminations. The construction is robust and produces a continuum of 2-filling rays in distinct mapping class group orbits.

For any orientable surface S of infinite type, we show any finite subgroup of its mapping class group can be realized as a group of isometries for some hyperbolic metric on S. We use this to classify torsion elements in certain big mapping class groups, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S. Finally, we show that compact subgroups of the mapping class group of S are finite, and locally compact subgroups are discrete.

We show a sharp lower bound 1/2 for the stable commutator length (scl) of any hyperbolic element in a group acting on a tree, provided that every edge stabilizer is relatively torsion-free in vertex stabilizers. The sharp bound becomes 1/2-1/n if the edge stabilizers are n-relatively torsion-free in vertex stabilizers. As an application, we show any element in a graph product of torsion-free groups has scl at least 1/2 unless it is supported in a complete subgraph. This in particular implies a sharp gap 1/2 for all right-angled Artin groups. We also characterize scl in vertex stabilizers, which allows us to show that any 3-manifold group has a spectral gap, that is a constant C such that any element g has either scl(g)>=C or scl(g)=0. We also classify elements whose scl vanish. The method is topological in nature. In many cases we also construct explicit quasimorphisms detecting these lower bounds, including a new short proof of the gap 1/2 in free groups.

Let G denote the mapping class group of the plane minus a Cantor set. We show that every action of G on the circle is either trivial or semi-conjugate to a unique minimal faithful action on the so-called simple circle. In the appendix, we compute the second homology of the mapping class group of the sphere minus a Cantor set.

Let G be a group acting on a tree with cyclic edge and vertex stabilizers. Then stable commutator length (scl) is rational in G. A large portion of the paper concentrates on the special case of Baumslag-Solitar groups, giving a criterion for existence of extremal surfaces, and proving that scl in so-call "surgery" families varies predictably and converges to scl in free groups. This is a homological analog of the phenomenon of geometric convergence in hyperbolic Dehn surgery.

We show that for a free product of torsion-free groups, scl(g) is at least 1/2 for any g not conjugate into any factor. Similar sharp bounds are obtained assuming the factors have no small torsion. The method is topological and new for scl estimates, and is pushed further to groups acting on trees in "Spectral gap of scl in graphs of groups and applications" with Nicolaus Heuer. The method turns out to be closely related to Anton Klyachko's car motion argument used in his elegant proof of the Kervaire conjecture for free products of torsion-free groups.

We show that scl is rational in free products of groups with vanishing scl. We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show that scl in free products of cyclic groups behaves in a piecewise quasi-rational way when the word is fixed but the orders of factors vary.