Question: What did authority collapse during the 7th century Greek peninsula?
About the article "Greece", section "Medieval period (4th century–1453)".

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Medieval period (4th century–1453).
The Roman Empire in the east, following the fall of the Empire in the west in the 5th century, is conventionally known as the Byzantine Empire (but was simply called "Roman Empire" in its own time) and lasted until 1453.
With its capital in Constantinople, its language and literary culture was Greek and its religion was predominantly Eastern Orthodox Christian.
From the 4th century, the Empire's Balkan territories, including Greece, suffered from the dislocation of the "Barbarian Invasions".
The raids and devastation of the Goths and Huns in the 4th and 5th centuries and the Slavic invasion of Greece in the 7th century resulted in a dramatic collapse in imperial authority in the Greek peninsula.
Following the Slavic invasion, the imperial government retained control of only the islands and coastal areas, particularly cities such as Athens, Corinth and Thessalonica, while some mountainous areas in the interior held out on their own and continued to recognize imperial authority.
Outside of these areas, a limited amount of Slavic settlement is generally thought to have occurred, although on a much smaller scale than previously thought.
The Byzantine recovery of lost provinces began toward the end of the 8th century and most of the Greek peninsula came under imperial control again, in stages, during the 9th century.
This process was facilitated by a large influx of Greeks from Sicily and Asia Minor to the Greek peninsula, while at the same time many Slavs were captured and re-settled in Asia Minor and those that remained were assimilated.
During the 11th and 12th centuries the return of stability resulted in the Greek peninsula benefiting from strong economic growth – much stronger than that of the Anatolian territories of the Empire.
Following the Fourth Crusade and the fall of Constantinople to the "Latins" in 1204 most of Greece quickly came under Frankish rule (initiating the period known as the "Frankokratia") or Venetian rule in the case of some of the islands.
The re-establishment of the Byzantine Empire in Constantinople in 1261 was accompanied by the recovery of much of the Greek peninsula, although the Frankish Principality of Achaea in the Peloponnese remained an important regional power into the 14th century, while the islands remained largely under Genoese and Venetian control.
In the 14th century much of the Greek peninsula was lost by the Empire as first the Serbs and then the Ottomans seized imperial territory.
By the beginning of the 15th century, the Ottoman advance meant that Byzantine territory in Greece was limited mainly to the Despotate of the Morea in the Peloponnese.
After the fall of Constantinople to the Ottomans in 1453, the Morea was the last remnant of the Byzantine Empire to hold out against the Ottomans.
However, this, too, fell to the Ottomans in 1460, completing the Ottoman conquest of mainland Greece.
With the Turkish conquest, many Byzantine Greek scholars, who up until then were largely responsible for preserving Classical Greek knowledge, fled to the West, taking with them a large body of literature and thereby significantly contributing to the Renaissance.
Slavic invasions and settlements.
From 27 BC until the end of the 6th century, the Peloponnese formed part of the province of Achaea, which during Late Antiquity comprised also the eastern parts of Central Greece.
Its capital was Corinth.
From the 580s on, as attested in the "Chronicle of Monemvasia", the Slavic invasions which affected the wider Balkans reached the peninsula, and led to the abandonment of the urban centres of Antiquity in favour of either remote or offshore locations like Monemvasia, while the inhabitants of several cities such as Patras are claimed by the "Chronicle" to have emigrated to Italy altogether.
Along with statements by medieval chroniclers like Isidore of Seville and Constantine Porphyrogennetos that Greece had completely submitted to the Slavs, this led to assumptions by earlier scholars that collapse of Byzantine control was total, and that the Peloponnese remained outside imperial control for two centuries afterwards.
The literary evidence is partially confirmed by several buried coin hoards from the 570s/580s and early 7th century, which attest to large-scale upheaval in two waves, one that peaked ca.
587, the date provided by the "Chronicle", and one that peaked in the far larger crisis of the reign of Heraclius (r. 610–641).
Literary, toponymic and archaeological evidence on the other hand shows that the Slavic invaders settled mostly in the western half of the peninsula, i.e.
the fertile plains of Elis and Messenia, Achaea and the plateau of Arcadia, while Byzantine authority survived in the more mountainous eastern parts of the peninsula as well as in various outposts around the coast, including Patras.
Nevertheless, as the rapid-re-Hellenization of the peninsula in the 9th century shows, a large Greek-speaking population must have remained in the areas overrun by the Slavs. 
During the period of Iberian civilization, the Catalan territory was home to several distinct tribes: The Indiketes in Empordà, the Ceretans in Cerdanya and the Airenosins in the Val d'Aran.
The influx of Celtic peoples led to a characteristic blend of cultures known as Celtiberian, which was affected by the first arrival of colonists from Ancient Greece and Carthage; like the rest of the Iberian Peninsula, Catalonia participated in what became the Iberian culture.
At this time Empúries (originally Greek "Emporion" market, then "Emporiae"), on the coast of what is now the Catalan province of Girona, a commercial enclave, founded from the Greek city of Phocaea in the 6th century BC.
From the 8th century BC to the 7th century BC the indigenous peoples came into contact with the colonizers, and the first iron objects are found in the area.
From the 7th century BC to the middle of the 5th century BC, the process of Iberianization was consolidated.
A period of plenty lasted from the middle of the 5th century BC until the 3rd century BC.
Finally, after the 218 BC arrival of the Romans, the Iberian culture was absorbed into that of Rome. 
The area remained part of the Byzantine empire until the late early 7th century when the Slavic migration destroyed Byzantine authority in most of the Balkan peninsula.
Since then Dardania ceased to exist as separate administrative entity. 
Thessaly remained part of the East Roman "Byzantine" Empire after the collapse of Roman power in the west, and subsequently suffered many invasions, such as by the Slavic tribe of the Belegezites in the 7th century AD.
The Avars had arrived in Europe in the late 550s.
They asserted their authority over many Slavs, who were divided into numerous petty tribes.
Many Slavs were galvanized into an effective infantry force, by the Avars.
In the 7th Century the Avar-Slav alliance began to raid the Byzantine Empire, laying siege to Thessaloniki and even the imperial capital Constantinople itself.
By the 8th Century Slavs had occupied most of the Balkans, from Austria to the Peloponnese, and from the Adriatic to the Black seas, with the exception of the coastal areas and certain mountainous regions of the Greek peninsula.
Far fewer numbers of Slavs appear to have settled in those parts of Greece where Slavs did settle.
Relations between the Slavs and Greeks were probably peaceful apart from the (supposed) initial settlement and intermittent uprisings.
Being agriculturalists, the Slavs probably traded with the Greeks inside towns.
It is likely that the re-Hellenization had already begun by way of this contact.
This process would be completed by a newly reinvigorated Byzantine Empire.
With the abatement of Arab-Byzantine Wars the Byzantine Empire began to consolidate its power in those areas of mainland Greece occupied by Proto-Slavic tribes.
Following the campaigns of the Byzantine general Staurakios in 782–783, the Byzantine Empire recovered Thessaly (then known as Hellas), taking many Slavs as prisoners.
Apart from military expeditions against Slavs, the re-Hellenization process begun under Nicephorus I involved (often forcible) transfer of peoples.
Many Slavs were moved to other parts of the empire, such as Anatolia and made to serve in the military.
In return, many Greeks from Sicily and Asia Minor were brought to the interior of Greece, to increase the number of defenders at the Emperor's disposal and dilute the concentration of Slavs.
Even non-Greeks were transferred to the Balkans, such as Armenians.
As more of the peripheral territories of the Byzantine Empire were lost in the following centuries, e.g.
Sicily, southern Italy and Asia Minor, their Greek-speakers made their own way back to Greece.
In 977 it was raided by the Bulgarians.