In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 elements in CAT(0) groups, and the set of pseudo-Anosov elements in mapping class groups. Regarding a proper action, the set of non-contracting elements is proven to be growth-negligible. In particular, for mapping class groups, the set of pseudo-Anosov elements is generic in a sufficiently large subgroup, provided that the subgroup has purely exponential growth. By Roblin's work, we obtain that the set of hyperbolic elements is generic in any discrete group action on CAT(-1) space with finite BMS measure. Applications to the number of conjugacy classes of non-contracting elements are given for non-rank-1 geodesics in CAT(0) groups with rank-1 elements.
This generalizes the recent result of Cumplido and Wiest @cite_24 in mapping class groups that a positive proportion of elements are pseudo-Anosov. See also Cumplido @cite_20 for a similar result in Artin-Tits groups.
@cite_24 @cite_20
2691309097 2952236103 2015253075 1964155237
Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that "most" elements of Artin-Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup @math of an Artin-Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin-Tits groups of spherical type, their pure subgroups and some of their commutator subgroups. In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group. In this paper we prove the results announced in [13]. Let G be a semi- simple, simply connected algebraic group defined over an algebraically closed field k. Let T be a maximal torus, B a Bore1 subgroup, B 3 T. Let W be the Weyl group of G. Let R (resp. R+ ) be the set of roots (resp. positive roots) relative to T (resp. B). Let S be the set of simple roots in R+. Let P be a maximal parabolic subgroup in G with associated fundamental weight w. Let W, be the Weyl group of P, and Wp be the set of minimal representatives of W W,. For w E Wp, let e(w) be the point and X(w) the Schubert variety in G P associated to w. In this paper we deter- mine the multiplicity m,(w) of X(w) at e(z), where e(z) E X(w), for all minuscule P’s and also for P = Pgn, G being of type C, (here Pun denotes the maximal parabolic subgroup obtained by omitting a,). The determina- tion of m,(w) is done as follows. Let L be the ample generator of Pic(G P). A basis has been constructed for @(X(w), L”) in terms of standard monomials on X(w) (cf. [ 16, 11 I). Let U; be the unipotent subgroup of G generated by U-,, ?ET(R+ - Rp+) (here R, denotes the set of roots of P and U, denotes the unipotent subgroup of G, associated to tl E R). Then U; e(r) gives an affme neighborhood of e(z) in G P. Let A, be the affine algebra of U, e(z) and A,.. = A, &, where & is the ideal of elements of A, that vanish on X(w) n U; e(z). Let M,,, be the maximal ideal in A, H, corresponding to e(r). Then using the results of [ 16, 111, we obtain a basis of M;, &f:,+,,’ . This enables us to obtain an inductive formula for F,,,, the Hilbert polynomial of X(w) at e(T) (cf. Corollaries 3.8 and 4.11), and also express m, (w) in terms of m, (w’)‘s, X(w’)‘s being the Schubert divisors in X(w) such that e(T)E X(w’) (cf. Theorems 3.7 and 4.10). Using this we We conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof the relations is (log|G|)O(1). We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name ofG, assuming in the case of Lie type simple groups overGF(pm) that an irreducible polynomialfof degreemoverGF(p) and a primitive root ofGF(pm) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groupsPSU(3, q) = 2A2(q), the Suzuki groupsSz(q) = 2B2(q), and the Ree groupsR(q) = 2G2(q). In particular,all finite groups G without composition factors of these types have presentations of length O((log|G|)3). For groups of Lie type (normal or twisted) of rank ≥ 2, we use a reduced version of the Curtis–Steinberg–Tits presentation.