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Let 
$G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let 
$N$ be an infinite normal subgroup of 
$G$ and let 
$\unicode[STIX]{x1D6FF}_{N}$ and 
$\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of 
$N$ and 
$G$ with respect to the pseudo-metric induced by the action. We prove that if 
$G$ has purely exponential growth with respect to the pseudo-metric, then 
$\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.
Let 
$X$ be an 
$n$-dimensional, finite, simply connected 
$\text{CW}$ complex and set
$${{\alpha }_{X}}\,=\,\underset{i}{\mathop{\lim \,\sup }}\,\frac{\log \,\text{rank}\,{{\pi }_{i}}\left( X \right)}{i}$$
When 
$0<{{\alpha }_{X}}<\infty $, we give upper and lower bounds for 
$\sum\nolimits_{i=k+2}^{k+n}{\,\text{rank}}\text{ }{{\pi }_{i}}\left( X \right)$ for 
$k$ sufficiently large. We also show for any 
$r$ that 
$\alpha x$ can be estimated from the integers 
$\text{rk }{{\pi }_{i}}\left( X \right),\,i\,\le \,nr$ with an error bound depending explicitly on 
$r$.
We consider d-dimensional stochastic processes 
which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some 
Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.
We consider d-dimensional stochastic processes 
which take values in (R+)d. These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some 
Here τ: (R+)d → R+, |x| = Σ1d |x(i)| A = {x ∈ (R+)d: |x| = 1} and T: A → A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Znρ−n = wp. This result is a generalization of the main limit theorem for super-critical branching processes; note, however, that in the present situation both p and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.
