The wikipedia definition of Absolute fitness is "the ratio between the number of individuals with that genotype after selection to those before selection. It is calculated for a single generation and must be calculated from absolute numbers."
But in one generation there are no reproductive events so it should be theoretically impossible for the population to be larger after the selection event than before it (since no reproduction is taking place).
That definition in the wikipedia is misleading at best. Contrary to what it seems to suggest there, absolute fitness is computed across a single generation, not within a single generation. For simplicity, let's look at an example using asexual haploids with discrete non-overlapping generations. If individuals with the A genotype produce an average of 3 offspring, each of which survive with probability 0.5, the A genotype has an absolute fitness of 1.5. The key is that we've multiplied fecundity (3) by viability (0.5) and this allows us to obtain a value greater than 1.
Things get more complicated if we look at diploid sexuals. The absolute fitness of the Aa genotype is not the ratio of the number of individuals with the Aa genotype at time t+1 to the number of Aa at time t. An easy way to see this: in a population of AA and aa parents only, e.g. as produced by a cross, the ratio of Aa at t+1 to Aa at t is infinite, which would imply infinite fitness if we went this with definition. Rather, the absolute fitness of the Aa genotype is one-half the number of surviving offspring at time t+1 produced by Aa parents at time t. The one-half comes from the fact that a parent individual contributes only one-half of the genetic material of each surviving offspring. For some traits, e.g. those involved in sex ratio determination or parental care, it will not be sufficient to count surviving offspring; in these cases we apply comparable logic to counting grand-offspring.
The wikipedia seems to suggest we are to look only at viability because we are comparing before-selection numbers to after-selection numbers, within a single generation and thus without an opportunity for fecundity to come into play. That is not consistent with the common usage of the term. To be fair, I have on rare occasion seen "absolute fitness" used as a synonym for "viability"--though I cannot think of any good justification for this usage.
That definition is problematic and incomplete. Absolute fitness can be used in different ways, and can be calculated at different levels of organisation (individuals, genes etc). However, absolute fitness always relate to actual growth rate, actual numbers of offspring or other measures of fitness (so an absolute measure), while relative fitness is calculated as a fitness ratio in relation to a specific genotype (so a relative measure).
Absolute fitness is often defined in terms of the per capita growth rate of a genotype (or the intrinsic rate of increase (r) for e.g. bacterial clones), which can clearly be above or below one. However, how to measure this is another thing, and this is where many different metrics are used. In quantitative genetics and life history evolution it is common to use the net reproductive rate (R0) derived from the characteristic equation as a measure of fitness. However, sometimes the number of offspring or other measures are used as proxies for absolute fitness (as fitness components).
The definition of fitness is a tricky subject, and what to use depends on exactly what type of organism you are studying. The Wikipedia definition is clearly inadequate, and I feel that the formulation "...is calculated for a single generation..." is a direct mistake. The section "Measures of fitness" in Roff (1992) and Day & Otto (2001) (chapter on fitness from the Encylopedia of Life Sciences) are both very useful to read to better understand different measures of fitness.
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Definition of Absolute Fitness
Let's forget about the wikipedia definition for a little while. There exists different definitions of fitness. They vary only slightly and usually it does not really matter. Let's talk only about absolute fitness at first. Absolute fitness can be calculated
fitness can be calculated for
Also, absolute fitness can be calculated in function of
Very often, when talking about theoretical concepts, we consider exclusively for semelparous species (species that have only one reproductive event and then die like some salmon species for example). In such case, when talking about one generation, we think of a population of adults that gets entirely replaced by a new population of offspring. Thinking about semelparous species has the advantage (among other advantages) to avoid confusion about lifetime reproductive success versus rate at which offsprings are produced. So, yes in one generation there is reproductive events. Even if selection on survival hasn't yet occurred on the offsrpings of the parents in a given generation, the fitness of these parents during this generation is calculated in function of the number of offspring that will survive later. Hope that makes sense.
Relating the above definition to your question
Now, let's relate the above to the wikipedia definition. Absolute fitness is NOT "the ratio between the number of individuals with that genotype after selection to those before selection". It is only the number of offsrpings produced by an individual for a given period (often the whole life of the individual. It can eventually be the number of offsprings that survive to a given age (selection on survival). Wikipedia seems to be wrong unless it has been misquoted. You should indicate where you read that so that one can correct wiki if there is indeed an error.
Relative Fitness
Now, relative fitness of a genotype/haplotype/individual for a given period of time (eventually the whole life of the individuals of interest) is just the absolute fitness of the individuals of interest divided by a standard absolute fitness. This standard absolute fitness is either the maximal absolute fitness observed in the population (or metapopulation) of interest or the maximum theoretical fitness based on some assumptions. For example we may calculate the relative fitness of an individual by dividing the absolute fitness of this individual by the fitness of a theoretical individual that would carry only the best possible mutations even though this individual doesn't exist in the population.
I think that if we want to settle on some definitions we should look into the book: Endler, J.A. (1986) Natural Selection in the Wild
Definition of Relative fitness
Relative fitness can also be measured with reference to a particular phenotype (or genotype), in which case $\overline {w}$ is not necessarily 1; this is the most common method used for polymorphic traits. If the population is sampled twice (or more) within a generation so that individuals in the second sample represent a subset of those sampled in the first sample (as in a capture-recapture or cohort study), then absolute fitnesses can be calculated. Examples are the probability of surviving between samples, or the probability of mating. On the other hand, if samples are made without replacement, or if samples are made of juveniles and adults at a single time, then only relative fitness can be calculaqted; information on total numbers and mean fitness is lost (see discussions in O'Donald 1971, Horns and Harrison 1970, and Manley 1974).)
Definition of Absolute fitness
Adaptedness and Adaptation. Adaptedness is the degree to which an organism is able to live and reproduce in a given set of environments: the state of being adapted (Dobzhansky 1968a,b). Adaptation is the process of becoming adapted or more adapted (ibid.). Unfortunately, adaptation.is also used in the sense of an adaptive trait (Lewontin 1978), confounding the end product with the process (see also Dunbar 1982). An adaptive trait is "an aspect of the developmental pattern of the organism surviving and reproducing" (Dobzhansky 1956, 1968a). There are problems in defining precisely what adaptednedd is so that it can be measured (Dobzhansky 1956, 1968a,b; Stern 1970; Lewontin 1978; Dunbar 1982). One solution is to define it in the sense of absolute (rather than relative) fitness (Table 2.1). In this case it can be measured by the average absolute lifetime contribution to the breeding population by a phenotype or a class of phenotypes. It thus becomes intimately related to the actual (R) or intrinsic $(r_{m})$ rate of increase, or "Malthusian parameter," and these have actually been used as measures of fitness for populations and species, though there are some problems (Fisher 1930; Dobzhansky 1968a,b; Dunbar 1982). Adaptedness has also been defined as the mean absolute fitness (Sober 1984). [...]
Defining Generation
A generation is the interval of time between birth of an organism and the birth of its offsprings (reference from "Biology ninth edition, Rave, Johnson, Mason, Losos and Singer).
So in my interpretation, yes there is a reproductive event in one generation. If some phenotypes or genotypes are producing more offsprings compared to other individuals, and that there is a correlation between their fitness and the trait that hey have, than yes, it is probable that the population could increase. Especially if they survive, and have offsprings, enough resources, etc.
