ESV Luke 21:28 Now when these things begin to take place, straighten up and raise your heads, because your redemption (ἀπολύτρωσις) is drawing near.”
What was drawing near?
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You have asked two questions - one about the word "apolutrosis" and the other about what was drawing near.
Every lexicon I consulted stated the following well-known facts: "apolutrosis" is a strengthened form of "lutrosis" (from the cognate root "luo" = I loose or release; see A Souter, Newman, Thayer, W E Vine, Analytical Lexicon by Friberg et al, Thayer, etc). Both mean almost the same thing - the act of granting freedom or release from slavery or bondage by paying a ransom. Thus, in English, the closest word would be "manumission". Hence, these words are correctly translated, liberation, a setting free, manumission, redemption, etc.
The word "apolutrosis" is used in a number of fascinating ways - it is used as an act in the past (an accomplished fact), a current process, and a future event. In all cases (except Heb 11:35 literal of actual slaves) it is used spiritually as a metaphor of Christ's salvation on our behalf.
The reference to something that was drawing near:
Jesus' statement about redemption/liberation drawing near Luke 21:28 is couched in terms of His instruction to the disciples to "look up" - in the previous verses he teaches them about the coming signs for which they are to watch. Specifically, they were to watch for two sets of "signs" or "omens" (Greek "semeia"), namely:
It is these (future) events that presage the final liberation/redemption of God's people.
I suspect you have already looked at the lexicon entries for the word, which supply definitions of setting free, deliverance, release, liberation, recovery, redemption.1. Another answer discussed the roots of the word ἀπολύτρωσις (apolutrōsis) as it relates to the verb λύω (luō) - meaning "to set free"2. The word λύτρον (lutron) - ransom - is also related.3 The word λύτρωσις (lutrōsis) by itself means essentially the same thing; the prefix ἀπο (apo) adds a sense of completeness.3
I believe the correct understanding of the word in context here is a description of liberation from (bodily) corruption rather than in terms of ransom. The preceding verse - And then they shall see the Son of Man coming in a cloud with power and great glory - clearly relates this event of redemption to the second coming of Christ. Romans 8:23, using the same word, also conveys this sense:
And not only they, but ourselves also, which have the firstfruits of the Spirit, even we ourselves groan within ourselves, waiting for the adoption, to wit, the redemption of our body.
The Byzantine (Greek) commentator Theophylact explained the Greek wording here in his commentary on Luke:
Just as the first coming of the Lord was for the re-fashioning and rebirth of our souls, so the second coming will be for the rebirth of our bodies. Death came first to the souls of Adam and Eve when they disobeyed, and nine hundred years after their disobedience their bodies underwent death in physical actuality. Consequently, our souls are reborn and made better by the first coming of Christ, and our bodies, by the second coming. Therefore the Lord says, "When these things come to pass, look up, you who are weighted down by corruption and you will be set free. For your redemption is at hand, that is, the perfect liberation of your soul and body together." The prefix apo intensifies the sense that the body at the second coming will be completely and totally delivered from corruption by the grace of the Lord Who abolishes the final enemy, death. First He overthrew the dark principalities and powers and redeemed our souls. But death remained, feeding upon our bodies. When the Lord abolishes death, it will result in our complete freedom and redemption.5
1. e.g., Newman, A Concise Greek-English dictionary of the New Testament; Swanson; Swanson, Dictionary of Biblical Languages with Semantic Domains: Greek (New Testament); The Lexham Analytical Lexicon to the Greek New Testament; Louw & Nida, Greek-English lexicon of the New Testament: Based on Semantic Domains
2. See, e.g., Newman
3. See, e.g., Lidell, An Intermediate Greek-English Lexicon
4. See, e.g., Royster, St. Paul's Epistle to the Romans: A Pastoral Commentary (St. Vladimir's Seminary Press, 2008), p.86
5. Explanation of the Gospel According to St. Luke (tr. from Greek; Chrysostom Press, 1997), p.275
You had a question about redemption in Luke 1:28, the answer is linked to the previous verse Luke 1:27 which tells of the timing and context of this event. The redemption being presented is that of the elect or the saved who will still be alive after the "Great Tribulation" Those who were "bought with a price" (1 Cor 6:20). Because these "saints" are still alive at that time, they have not yet been delivered to their rightful owner Jesus Christ. At the second coming of Jesus Christ He will then collect those of His that are still alive, those who are marked with the Holy Spirit (Eph 1:13). These saints had been bought and are owned but had not yet been delivered. The delivery of the saints is at death. The reason is that only spiritual beings can enter and reside in heaven. Believers belong to Christ while we are alive on earth, but we are redeemed and delivered to our owner after our physical death. I hope this helps.
Jesus' words may be an allusion to Daniel 4:33 from the LXX.
In the Hebrew it has Nebuchadnezzar "lifting up his eyes to heaven". We are probably to take this as him repenting of his pride and saving his kingdom by his repentance:
KJV Daniel 4:34 And at the end of the days I Nebuchadnezzar lifted up mine eyes unto heaven, and mine understanding returned unto me, and I blessed the most High, and I praised and honoured him that liveth for ever, whose dominion is an everlasting dominion, and his kingdom is from generation to generation:
In the LES the king doesn't lift his eyes to heaven but the king says "my time of recovery [ἀπολύτρωσις] came":
Lexham English Septuagint Daniel 4:34 30c And at the completion of the seven years, my time of recovery [ἀπολύτρωσις] came, and my sins and acts of my ignorance were brought to an end before the God of heaven, and I prayed concerning my sins against the great God of gods. And behold, a messenger addressed me from heaven, saying, “O Nebuchadnezzar, serve the holy God of heaven and give honor to the most high, for the royal dwelling of your nation is being given back to you.”
Brannan, R., Penner, K. M., Loken, I., Aubrey, M., & Hoogendyk, I. (Eds.). (2012). The Lexham English Septuagint (Da 4:30c–33). Bellingham, WA: Lexham Press.
Swete's LXX Daniel 4:33 ἕως δὲ πρωὶ πάντα τελεσθήσεται ἐπὶ σέ, Ναβουχοδονοσὸρ βασιλεῦ Βαβυλῶνος, καὶ οὐχ ὑστερήσει ἀπὸ πάντων τούτων οὐθέν.  Ἐγὼ Ναβουχοδονοσόρ, βασιλεὺς Βαβυλῶνος, ἑπτὰ ἔτη ἐπεδήθην· χόρτον ὡς βοῦν ἐψώμισάν με καὶ ἀπὸ τῆς χλόης τῆς γῆς ἤσθιον. καὶ μετὰ ἔτη ἑπτὰ ἔδωκα τὴν ψυχήν μου εἰς δέησιν, καὶ ἠξίωσα περὶ τῶν ἁμαρτιῶν μου κατὰ πρόσωπον Κυρίου τοῦ θεοῦ τοῦ οὐρανοῦ, καὶ περὶ τῶν ἀγνοιῶν μου τοῦ θεοῦ τῶν θεῶν τοῦ μεγάλου ἐδεήθην.  καὶ αἱ τρίχες μου ἐγένοντο ὡς πτέρυγες ἀετοῦ, οἱ ὄνυχές μου ὡσεὶ λέοντος· ἠλλοιώθη ἡ σάρξ μου καὶ ἡ καρδία μου, γυμνὸς περιεπάτουν μετὰ τῶν θηρίων τῆς γῆς. ἐνύπνιον εἶδον καὶ ὑπόνοιαὶ με εἰλήφασιν, καὶ διὰ χρόνου ὕπνος με ἔλαβε πολὺς καὶ νυσταγμὸς ἐπέπεσέ μοι.  καὶ ἐπὶ συντελείᾳ τῶν ἐπτὰ ἐτῶν ὁ χρόνος μου τῆς ἀπολυτρώσεως ἦλθεν, καὶ αἱ ἁμαρτίαι μου καὶ αἱ ἄγνοιαί μου ἐπληρώθησαν ἐναντίον τοῦ θεοῦ τοῦ οὐρανοῦ· καὶ ἐδεήθην περὶ τῶν ἀγνοιῶν μου τοῦ θεοῦ τῶν θεῶν τοῦ μεγάλου, καὶ ἰδοὺ ἄγγελος εἷς ἐκάλεσέν με ἐκ τοῦ οὐρανοῦ λέγων Ναβουχοδονοσόρ, δούλευσον τῷ θεῷ τοῦ οὐρανοῦ τῷ ἁγίῳ καὶ δὸς δόξαν τῷ ὑψίστῳ· τὸ βασίλειον τοῦ ἔθνους σού σοι ἀποδίδοται.
If this is what Jesus/Luke was alluding to then he wants to call attention to the fact that the Jews were to look up to await their "recovery" just as Nebuchadnezzar's "recovery" came and he saw his sins and acts of ignorance for what they were and committed to serve the God of heaven.
This might be a reference to:
KJV Zechariah 12:10: And I will pour upon the house of David, and upon the inhabitants of Jerusalem, the spirit of grace and of supplications: and they shall look upon me whom they have pierced, and they shall mourn for him, as one mourneth for his only son, and shall be in bitterness for him, as one that is in bitterness for his firstborn.
It may also refer to this:
[Isa 62:11-12 KJV] 11 Behold, the LORD hath proclaimed unto the end of the world, Say ye to the daughter of Zion, Behold, thy salvation cometh; behold, his reward [is] with him, and his work before him. 12 And they shall call them, The holy people, The redeemed of the LORD: and thou shalt be called, Sought out, A city not forsaken.
They would also have had good reason to expect the dramatic intervention in Zechariah 14:
[Zec 14:1-21 KJV] 1 Behold, the day of the LORD cometh, and thy spoil shall be divided in the midst of thee. 2 For I will gather all nations against Jerusalem to battle; and the city shall be taken, and the houses rifled, and the women ravished; and half of the city shall go forth into captivity, and the residue of the people shall not be cut off from the city. 3 Then shall the LORD go forth, and fight against those nations, as when he fought in the day of battle. 4 And his feet shall stand in that day upon the mount of Olives, which [is] before Jerusalem on the east, and the mount of Olives shall cleave in the midst thereof toward the east and toward the west, [and there shall be] a very great valley; and half of the mountain shall remove toward the north, and half of it toward the south. 5 And ye shall flee [to] the valley of the mountains; for the valley of the mountains shall reach unto Azal: yea, ye shall flee, like as ye fled from before the earthquake in the days of Uzziah king of Judah: and the LORD my God shall come, [and] all the saints with thee. 6 And it shall come to pass in that day, [that] the light shall not be clear, [nor] dark: 7 But it shall be one day which shall be known to the LORD, not day, nor night: but it shall come to pass, [that] at evening time it shall be light. 8 And it shall be in that day, [that] living waters shall go out from Jerusalem; half of them toward the former sea, and half of them toward the hinder sea: in summer and in winter shall it be. 9 And the LORD shall be king over all the earth: in that day shall there be one LORD, and his name one. 10 All the land shall be turned as a plain from Geba to Rimmon south of Jerusalem: and it shall be lifted up, and inhabited in her place, from Benjamin's gate unto the place of the first gate, unto the corner gate, and [from] the tower of Hananeel unto the king's winepresses. 11 And [men] shall dwell in it, and there shall be no more utter destruction; but Jerusalem shall be safely inhabited. 12 And this shall be the plague wherewith the LORD will smite all the people that have fought against Jerusalem; Their flesh shall consume away while they stand upon their feet, and their eyes shall consume away in their holes, and their tongue shall consume away in their mouth. 13 And it shall come to pass in that day, [that] a great tumult from the LORD shall be among them; and they shall lay hold every one on the hand of his neighbour, and his hand shall rise up against the hand of his neighbour. 14 And Judah also shall fight at Jerusalem; and the wealth of all the heathen round about shall be gathered together, gold, and silver, and apparel, in great abundance. 15 And so shall be the plague of the horse, of the mule, of the camel, and of the ass, and of all the beasts that shall be in these tents, as this plague. 16 And it shall come to pass, [that] every one that is left of all the nations which came against Jerusalem shall even go up from year to year to worship the King, the LORD of hosts, and to keep the feast of tabernacles. 17 And it shall be, [that] whoso will not come up of [all] the families of the earth unto Jerusalem to worship the King, the LORD of hosts, even upon them shall be no rain. 18 And if the family of Egypt go not up, and come not, that [have] no [rain]; there shall be the plague, wherewith the LORD will smite the heathen that come not up to keep the feast of tabernacles. 19 This shall be the punishment of Egypt, and the punishment of all nations that come not up to keep the feast of tabernacles. 20 In that day shall there be upon the bells of the horses, HOLINESS UNTO THE LORD; and the pots in the LORD'S house shall be like the bowls before the altar. 21 Yea, every pot in Jerusalem and in Judah shall be holiness unto the LORD of hosts: and all they that sacrifice shall come and take of them, and seethe therein: and in that day there shall be no more the Canaanite in the house of the LORD of hosts.
Of course, we know that Jerusalem was razed, not raised because they did not receive their messiah so instead of glory they experienced severe judgment. Some see the release/rescue/redemption as occurring spiritually through the gospel instead. Others see those promises as still future.
I see them as having been offered in the first century by Jesus but refused. This release never happened yet. I'm still figuring out when and how such promises are going to be fulfilled, if at all.
The first spring equinox of the new age was drawing near. This is evidenced by the next verses of which one states that Pesach (Easter) was coming. Easter is calculated from the Spring equinox concerning the moon cycle.
Redemption and salvation are in relation to light and darkness. The spring equinox is not only a dawn where the light brings ‘redemption’ after the night, but also brings longer days and shorter nights; a double ‘victory’ over darkness.
When also the contemporary moon cycle comes to an end, it can be considered a calendrical recalibration, which is also evidenced in the calendrical etiological interpretations of Ragnarok (Rechne-Rück = Calculate back) and Armageddon (from Hor-m-Akhet On = Horus (day) M (in) Akhet (the horizon) On (Ancient Egyptian name for Heliopolis, which is the city of the sun).
Such an astronomical clockwork moment would have Great effects on the religious interpretations and Lucas saying to look up at the sky while Jesus was hugely mythologised by Solar cult elements, gives enough reasons to assume a special spring equinox was meant. From Aries to Pisces (the New Kingdom of heaven = the Age of Pisces.
Pisces being the follow up constellation of Aries has a symbolic interpretation of ‘lamb’ (sprout of Aries) of God.
In short redemption means ‘being in the light’ or in Ancient Egyptian terms; ‘going forth by day’. It refers to the renewal of life that the spring equinox brings with the return of the sun (son) in heaven. Jesus was also an astronomer. He hints to the age of Aquarius when his disciples ask him when he returns. Also in Luke (22-9 & 22-10). Also Luke speaks here of Jerusalem which functions as On (Heliopolis) with mount Armageddon (Tel Megiddo) functioning as Hor-m-Akhet On. Interestingly the Hormakhet name is ascribed to the Great Sphinx which is looking due east towards the... spring equinox.
I will edit the etymological part of the Greek word soon. This is a work in progress 😃