Bounding the homological finiteness length

Gandini, Giovanni

doi:10.1112/blms/bds047

Cited by 17 publications

(19 citation statements)

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“…The problem of determining the finiteness length of an -arithmetic group is an ongoing challenge; see, for instance, the introductions of [22,24,26] In the arithmetic set-up, we combine Theorem 1.1 with results due to M. Bestvina, A. Eskin and K. Wortman [11] and G. Gandini [36] to obtain the following proof of (a generalization of) Bux's equality. Theorem 1.3 Let P be a proper parabolic subgroup of a non-commutative, connected, reductive, split linear algebraic group G defined over a global field K. Denote by U P the unipotent radical of P and by T P a maximal torus of G contained in P. For any -arithmetic subgroup Γ ≤ U P T P , the following inequalities hold.…”

Section: Theorem -Motivation and Examplesmentioning

confidence: 99%

On the Finiteness length of some soluble linear groups

Rego

2021

Can. J. Math.-J. Can. Math.

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Given a commutative unital ring , we show that the finiteness length of a group is bounded above by the finiteness length of the Borel subgroup of rank one B • 2 ( ) = * * 0 * ≤ SL 2 ( ) whenever admits certain -representations with metabelian image. Combined with results due to Bestvina-Eskin-Wortman and Gandini, this gives a new proof of (a generalization of) Bux's equality on the finiteness length of -arithmetic Borel groups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels' groups A ( ) ≤ GL ( ) in terms of and B • 2 ( ). This generalizes earlier results due to Remeslennikov, Holz, Lyul'ko, Cornulier-Tessera, and points out to a conjecture about the finiteness length of such groups.The purpose of this paper is twofold. First, using algebraic methods, we give an upper bound on the finiteness length of groups with certain soluble -representations, including many parabolic subgroups of classical groups. Second, we record the current state of knowledge on the finiteness lengths of H. Abels' soluble matrix groups {A ( )} ≥2 . This includes a different, topological proof of an unpublished result of R. Strebel, who classified which A ( ) are finitely presented. Our main results are the following.

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Section: Theorem -Motivation and Examplesmentioning

confidence: 99%

On the Finiteness length of some soluble linear groups

Rego

2021

Can. J. Math.-J. Can. Math.

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“…Very little is known about higher finiteness properties for lattices in hyperbolic buildings, apart from a recent result of Gandini [29] which bounds the homological finiteness length of non-cocompact lattices in Aut(X) for X a locally finite contractible polyhedral complex. As a corollary, such lattices are not finitely presentable.…”

Section: Property (T) and Finiteness Properties Ballmann-świ ֒mentioning

confidence: 99%

Lattices in hyperbolic buildings

Thomas¹

2016

Geometry, Topology, and Dynamics in Negative Curvature

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1.1. Coxeter groups and Coxeter polytopes. We mostly follow the reference Davis [19], particularly Chapter 6, and concentrate on the hyperbolic case.Recall that a Coxeter group is a group W with finite generating set S and presentation of the form W = s ∈ S | (st) mst = 1 where s, t ∈ S, m ss = 1 for all s ∈ S and if s = t then m st is an integer ≥ 2 or m st = ∞, meaning that the product st has infinite order. The pair (W, S) is called a Coxeter system. A Coxeter system (W, S) is right-angled if for each s, t ∈ S with s = t, m st ∈ {2, ∞}. Note that m st = 2 if and only if st = ts. Let X n be the n-dimensional sphere, n-dimensional Euclidean space or n-dimensional (real) hyperbolic space. Many important examples of Coxeter groups arise as discrete reflection groups acting on X n , as follows. Let P be a convex polyhedron in X n with all dihedral angles integer submultiples of π. Such a P is called a Coxeter polytope. Let W = W (P ) be the group generated by the set S = S(P ) of reflections in the codimension one faces of P . Then (W, S) is a Coxeter system and W is a discrete subgroup of the isometry group of X n (see [19, Theorem 6.4.3]). Moreover, the action of W tessellates X n by copies of P . For example, let P be a right-angled hyperbolic p-gon, p ≥ 5. Then the corresponding Coxeter system is right-angled with p generators, one for each side of P , so that if s and t are reflections in distinct sides then m st = 2 when these sides are adjacent, and otherwise m st = ∞.Suppose that X n is the sphere or Euclidean space. Then Coxeter polytopes P ⊂ X n exist and have been classified in every dimension, and the corresponding Coxeter systems (W, S) are the spherical or affine Coxeter systems, respectively (see [19, Table 6.1] for the classification).If X n is n-dimensional hyperbolic space H n , then there is no complete classification of Coxeter polytopes. Vinberg's Theorem [65] establishes that compact hyperbolic Coxeter polytopes can exist only in dimension n ≤ 29, although at the time of writing the highest dimension in which an example is known is n = 8 (due to Bugaenko [12]). Finite volume hyperbolic Coxeter polytopes have also been investigated, with for example Prokhorov [50] proving these can exist only in dimension n ≤ 995. For n ≤ 6 (respectively, n ≤ 19), there are infinitely many essentially distinct compact (respectively, finite volume) hyperbolic Coxeter polytopes (Allcock [2]). In dimension 3, Andreev's Theorem [3] classifies compact hyperbolic Coxeter polytopes, but in dimensions n ≥ 4 only special cases have been considered, and there seems little hope of a complete list. Hyperbolic Coxeter polytopes which are simplices exist in dimensions n ≤ 4 only, and their classification is given in [19, Table 6.2]. Right-angled compact hyperbolic polytopes also exist in dimensions n ≤ 4 only, and there are infinitely many examples in each dimension n ≤ 4 (see [66]). A right-angled example in dimension 3 is the dodecahedron, which tessellates H 3 as depicted on the cover of Thurston's book [59], and a right-...

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“…In the arithmetic set-up, we combine Theorem A with results due to M. Bestvina, A. Eskin and K. Wortman [11], and G. Gandini [35] to obtain the following proof of (a generalization of) Bux's equality.…”

Section: Introductionmentioning

confidence: 99%

“…[48]. Since the stabilizers of this action are finite [19,Section 3.3], it follows that B • 2 (O S ) belongs to P. Kropholler's HF class and Gandini's theorem [35] applies, yielding φ(B • 2 (O S )) ≤ |S| − 1. An alternative proof of the inequality φ(Γ) ≤ |S|−1 above was announced by K. Wortman; see [11, p. 2169].…”

Section: Introductionmentioning

confidence: 99%

“…The reader familiar with the theory of S-arithmetic groups might notice that Theorem C is 'uninteresting' in characteristic zero-in this case, φ(Γ) does not depend on the cardinality of S by the Kneser-Tiemeyer localglobal principle [52,Theorem 3.1], and moreover that φ(Γ) = ∞ by [52,Theorem 4.3 and (the proof of) Theorem 4.4]. Nevertheless, the charm of Theorem C lies in the independency of characteristic and in the content of the three theorems used to prove it, namely: higher dimensional isoperimetric inequalities for S-arithmetic lattices [11], a homotopical obstruction intrinsic to the group schemes considered (Theorem A), and-in positive characteristic-a geometric obstruction that occurs for many groups acting nicely on finite-dimensional contractible complexes [35].…”

Section: Introductionmentioning

confidence: 99%

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On the finiteness length of some soluble linear groups

Rego

2019

Preprint

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Let R be a commutative ring with unity. We prove that the finiteness length of a group G is bounded above by the finiteness length of the Borel subgroup of rank one B • 2 (R) = ( * * 0 * ) ≤ SL2(R) whenever G admits certain representations over R with soluble image. We combine this with results due to Bestvina-Eskin-Wortman and Gandini to give a new proof of (a generalization of) Bux's equality on the finiteness length of S-arithmetic Borel subgroups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing-in terms of n and B • 2 (R)-the finite presentability of Abels' soluble groups An(R) ≤ GLn(R). This generalizes earlier results due to Remeslennikov, Holz, Lyul'ko, Cornulier-Tessera, and points out to a conjecture about the finiteness lengths of such groups.

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Bounding the homological finiteness length

Cited by 17 publications

References 17 publications

On the Finiteness length of some soluble linear groups

On the Finiteness length of some soluble linear groups

Lattices in hyperbolic buildings

On the finiteness length of some soluble linear groups

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