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Register a new userEmbedding relatively hyperbolic groups in products of trees
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We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3-manifolds with non-empty boundary have asymptotic dimension 2.
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We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
We study cocompact lattices with dense projections in a product $G_1 times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only finitely many discrete subgroups of $G_1 times G_2$.
We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.
If $G$ is a free product of finite groups, let $Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti{e}rrez-Krsti{c} [M. Guti{e}rrez and S. Krsti{c}, {em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krsti{c} [W. Bogley and S. Krsti{c}, {em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $underline{E} Sigma Aut_1(G)$-space for these groups.
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