We prove that all maximal-tb Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m),(3,3),(3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances which induce faithful actions of the modular group PSL(2,Z) and the mapping class group M(0,4) into the coordinate rings of algebraic varieties associated to Legendrian links. Our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

Infinitely many Lagrangian fillings

Invariable generation of Thompson groups

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated.

On the complexity of the word problem for automaton semigroups and automaton groups

We complete the computation of growth rate of cluster algebras. In particular, we show that growth of all exceptional non-affine mutation-finite cluster algebras is exponential.

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Growth rate of cluster algebras

An unambiguous construction of an embedding of the free product of n cyclic groups of order p into the the group of finite state infinite unitriangular matrices over the finite field of size p is presented.

Representations of free products by infinite unitriangular matrices over finite fields

The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's rel...

Andrij Olijnyk

Papers

Representations of free products by infinite unitriangular matrices over finite fields

Inverse semigroups of partial automaton permutations