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CONFORMAL DIMENSION AND RANDOM GROUPS
"... Abstract. We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where l is the relator length, going to inf ..."
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Abstract. We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where l is the relator length, going to infinity. (a) 1 + 1/C < Cdim(∂∞G) < Cl / log(l), for the few relator model, and (b) 1 + l/(C log(l)) < Cdim(∂∞G) < Cl, for the density model, at densities d < 1/16. In particular, for the density model at densities d < 1/16, as the relator length l goes to infinity, the random groups will pass through infinitely many different quasiisometry classes. 1.
QUASICIRCLES THROUGH PRESCRIBED POINTS
"... Abstract. We show that in an Lannularly linearly connected, Ndoubling, complete metric space, any n points lie on a λquasicircle, where λ depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any virtually cyclic subgroup, then any geodesi ..."
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Abstract. We show that in an Lannularly linearly connected, Ndoubling, complete metric space, any n points lie on a λquasicircle, where λ depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any virtually cyclic subgroup, then any geodesic line in G lies in a quasiisometrically embedded copy of H2. 1.
CONFORMAL DIMENSION VIA SUBCOMPLEXES FOR SMALL CANCELLATION AND RANDOM GROUPS
"... Abstract. We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2 + o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like lK in the length l of the relators, then ..."
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Abstract. We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2 + o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like lK in the length l of the relators, then a.a.s. such a random group has conformal dimension 2 + K + o(1). In Gromov’s density model, a random group at density d < 1 8 a.a.s. has conformal dimension dl/  log d. The upper bound for C′ ( 1 8) groups has two main ingredients: `pcohomology (following Bourdon–Kleiner), and walls in the Cayley complex (building on Wise and Ollivier–Wise). To find lower bounds we refine the methods of [Mac12] to create larger ‘round trees ’ in the Cayley complex of such groups. As a corollary, in the density model at d < 1 8, the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group. 1.
3 SOME APPLICATIONS OF ℓpCOHOMOLOGY TO BOUNDARIES OF GROMOV HYPERBOLIC SPACES
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HAUSDORFF DIMENSION OF WIGGLY METRIC SPACES
"... ABSTRACT. For a compact connected set X ⊆ `∞, we define a quantity β′(x, r) that measures how close X may be approximated in a ball B(x, r) by a geodesic curve. We then show there is c> 0 so that if β′(x, r)> β> 0 for all x ∈ X and r < r0, then dimX> 1+cβ2. This generalizes a theorem ..."
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ABSTRACT. For a compact connected set X ⊆ `∞, we define a quantity β′(x, r) that measures how close X may be approximated in a ball B(x, r) by a geodesic curve. We then show there is c> 0 so that if β′(x, r)> β> 0 for all x ∈ X and r < r0, then dimX> 1+cβ2. This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.