Let G be a reductive group over the field F = k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne-Lusztig varieties Xx(b), which are indexed by elements b ∈ G(F ) and x ∈ W , were introduced by Rapoport [Rap00]. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman [GHKR10]. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for G(F ), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne-Lusztig set. We present two immediate applications of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.
Abstract. Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex. IntroductionThe divergence of a pair of geodesics is a classical notion related to curvature. Roughly speaking, given a pair of geodesic rays emanating from a basepoint, their divergence measures, as a function of r, the length of a shortest "avoidant" path connecting their time-r points. A path is avoidant if it stays at least distance r away from the basepoint. In [15], Gersten used this idea to define a quasi-isometry invariant of spaces, also called divergence. We recall the definitions of both notions of divergence in Section 2.The divergence of every pair of geodesics in Euclidean space is a linear function, and it follows from Gersten's definition that any group quasi-isometric to Euclidean space has linear divergence. In a δ-hyperbolic space, any pair of non-asymptotic rays diverges exponentially; thus the divergence of any hyperbolic group is exponential. In symmetric spaces of non-compact type, the divergence is either linear or exponential, and Gromov suggested in [16] the same should be true in CAT(0) spaces.Divergence has been investigated for many important groups and spaces, and contrary to Gromov's expectation, quadratic divergence is common. Gersten first exhibited quadratic divergence for certain CAT(0) spaces in [15]. He then proved in [14] that the divergence of the fundamental group of a closed geometric 3-manifold is either linear, quadratic or exponential, and characterised the (geometric) ones with quadratic divergence as the fundamental groups of graph manifolds. showed that all graph manifold groups have quadratic divergence. More recently, Duchin-Rafi [13] established that the divergence of Teichmüller space and the mapping class group is quadratic (for mapping class groups this was also obtained by Behrstock in [5]). Druţu-Mozes-Sapir [12] have conjectured that the divergence of lattices in higher rank semisimple Lie groups is always linear, and proved this conjecture in some cases. Abrams et al [1] and independently Behrstock-Charney [2] have shown that if A Γ is the right-angled Artin group associated to a graph Γ, the group A Γ has either linear or quadratic divergence, and its divergence is linear if and only if Γ is (the 1-skeleton of) a join.In this work we study the divergence of 2-dimensional right-angled Coxeter groups. Our first main result is Theorem 1.1 below, which characterises such groups with linear and quadratic divergence in terms of their defining graphs. This result can be seen as a step in the quasi-isometry classification of (right-angled) Coxeter groups, about which very little is known.We note that by [10], every right-angled Artin group is a finite index subgroup of, and therefore quasi-isometric to, a right-angled Coxeter group. However, in contrast to the setting of right-angled Artin gr...
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian [Bowditch, Acta Math. 180 (1998) 145-186]. Our main result gives an explicit, computable 'visual' construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the cocompact Fuchsian groups form a rigid quasi-isometry class. In Appendix B, written jointly with Christopher Cashen, we show that the JSJ tree is not a complete quasi-isometry invariant for the entire class of Coxeter groups we consider.We prove Theorem 1.2 in Section 3, and an explicit statement of this theorem appears as Theorem 3.37. A consequence of our description is Corollary 1.3. For Γ satisfying Standing Assumptions 1.1, there exists an algorithm to compute the JSJ tree of the right-angled Coxeter group W Γ .Corollary 1.5. The quasi-isometry problem is decidable for the class of right-angled Coxeter groups W Γ with Γ ∈ G.As a consequence of Bowditch's construction of the JSJ tree and Theorem 1.4, we also obtain that the visual boundary is a complete invariant for the same class of groups.Corollary 1.6. Let Γ, Λ ∈ G (where G is as defined above the statement of Theorem 1.4). Then W Γ and W Λ are quasi-isometric if and only if ∂W Γ and ∂W Λ are homeomorphic.The visual boundaries of hyperbolic groups have been investigated byŚwiątkowski and coauthors in a series of papers [31,[39][40][41][42]. In particular,in [31] Martin andŚwiątkowski consider visual boundaries of fundamental groups of graphs of groups with finite edge groups. Suppose now that W Γ and W Λ are 2-dimensional hyperbolic right-angled Coxeter groups with infinitely many ends. When applied to the above-mentioned tree-of-groups decompositions for W Γ and W Λ , the results of [31] imply that ∂W Γ and ∂W Λ are homeomorphic if and only if the corresponding sets of homeomorphism types of visual boundaries of 1-ended vertex groups are the same. Our results thus have applications to determining the homeomorphism type of ∂W Γ when W Γ has infinitely many ends.The work of Tukia [44], Gabai [22] and Casson-Jungreis [15] shows that the cocompact Fuchsian groups form a rigid quasi-isometry class, that is, any finitely generated group which is quasi-isometric to a Fuchsian group is cocompact Fuchsian. Our last application gives a direct proof of this fact for 2-dimensional right-angled Coxeter groups.Theorem 1.7. Any 2-dimensional right-angled Coxeter group which is quasi-isometric to a cocompact Fuchsian group is cocompact Fuchsian.We give a precise statement as Theorem 4.2 and prove this result in Section 4.2.We now discuss several quasi-isometry invariants that have previously been considered for right-angled Coxeter groups. In...
We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized Θ-graphs and cycles of generalized Θ-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter ≤ 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the right-angled Coxeter groups we consider, such that covers corresponding to torsionfree, finite-index subgroups are surface amalgams.
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