9,562 research outputs found
Bounding the homological finiteness length
We give a criterion for bounding the homological finiteness length of certain
HF-groups. This is used in two distinct contexts. Firstly, the homological
finiteness length of a non-uniform lattice on a locally finite n-dimensional
contractible CW-complex is less than n. In dimension two it solves a conjecture
of Farb, Hruska and Thomas. As another corollary, we obtain an upper bound for
the homological finiteness length of arithmetic groups over function fields.
This gives an easier proof of a result of Bux and Wortman that solved a
long-standing conjecture. Secondly, the criterion is applied to integer
polynomial points of simple groups over number fields, obtaining bounds
established in earlier works of Bux, Mohammadi and Wortman, as well as new
bounds. Moreover, this verifes a conjecture of Mohammadi and Wortman.Comment: Revised versio
Orbits of strongly solvable spherical subgroups on the flag variety
Let G be a connected reductive complex algebraic group and B a Borel subgroup
of G. We consider a subgroup H of B acting with finitely many orbits on the
flag variety G/B, and we classify the H-orbits in G/B in terms of suitable
combinatorial invariants. As well, we study the Weyl group action defined by
Knop on the set of H-orbits in G/B, and we give a combinatorial model for this
action in terms of weight polytopes.Comment: v4: final version, to appear on Journal of Algebraic Combinatorics.
Apported some minor corrections to the previous versio
Homological stability for automorphism groups of RAAGs
We show that the homology of the automorphism group of a right-angled Artin
group stabilizes under taking products with any right-angled Artin group.Comment: final versio
The Farrell-Jones conjecture for fundamental groups of graphs of abelian groups
We show that the Farrell-Jones Conjecture holds for fundamental groups of
graphs of groups with abelian vertex groups. As a special case, this shows that
the conjecture holds for generalized Baumslag-Solitar groups
Projective normality of model varieties and related results
We prove that the multiplication of sections of globally generated line
bundles on a model wonderful variety M of simply connected type is always
surjective. This follows by a general argument which works for every wonderful
variety and reduces the study of the surjectivity for every couple of globally
generated line bundles to a finite number of cases. As a consequence, the cone
defined by a complete linear system over M or over a closed G-stable subvariety
of M is normal. We apply these results to the study of the normality of the
compactifications of model varieties in simple projective spaces and of the
closures of the spherical nilpotent orbits. Then we focus on a particular case
proving two specific conjectures of Adams, Huang and Vogan on an analogue of
the model orbit of the group of type E8.Comment: v2: 54 pages, new introduction and several minor changes, added
Proposition 9.2. To appear on Representation Theor
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