Publications (69)
Hyperbolic buildings, affine buildings and automatic groups
Donald I. Cartwright, Michael Shapiro
We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely generated group acting co-compactly on such a building is hyperbolic, hence automat…
Simply-laced Coxeter groups and groups generated by symplectic transvections
Boris Shapiro, Michael Shapiro, Alek Vainshtein +1
Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u…
Growth in higher Baumslag-Solitar groups
Ayla P. Sánchez, Michael Shapiro
We study the HNN extension of given by the cubing endomorphism , and prove that such groups have rational growth. To do so, we describe a method of com…
Cluster structures on simple complex Lie groups and the Belavin-Drinfeld classification
Michael Gekhtman, Michael Shapiro, Alek Vainshtein
We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to…
Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective
Michael Gekhtman, Michael Shapiro, Alek Vainshtein
We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networ…
Cluster algebras of finite mutation type via unfoldings
Anna Felikson, Michael Shapiro, Pavel Tumarkin
We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every…